on the defectiveness of projective varieties/menu/standard/...of projective varieties, explaining...

FACOLTA DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di Laurea Magistrale in Matematica

On the defectiveness of projectivevarieties

Alessandro Oneto

Relatore:

Chiar.ma Prof.ssa

Maria Virginia Catalisano

Correlatore:

Chiar.mo Prof.

Enrico Carlini

Anno Accademico 2011-2012

Introduction

In 1770 Waring stated that every natural number can be written as sum of at

most 9 cubs of positive integers, or, more in general, that for any given degree

d there exists a number such that each positive integer can be written as sum

of at most g(d) dth-powers of natural numbers.

Such Number Theory problem can be translated as a problem dealing with

the decomposition of homogeneous polynomials in sum of powers of linear

forms. The minimal number of powers of linear forms needed to a given form

F to be written as sum of them is called the Waring rank, or simply the rank,

of F . Since the Veronese varieties V d,n parametrize the dth-powers of linear

forms in n+ 1 variables, if the sth-secant varieties fills the ambient space, then

a general form in n+1 variables is a sum of s powers of linear forms. Hence the

problem above became quickly an issue about secant varieties and defectiveness

of Veronese varieties.

In 1851 Sylvester found the equations of the sth-secant varieties to the

Rational Normal Curve, i.e. the Veronese variety V 1,d. He proved that such

equations are given by minors of the so-called catalecticant matrices and he

described an algorithm to found the canonical form of a general binary form

of odd degree 2m + 1, namely how to write such general binary form as sum

of m + 1 powers of linear forms. Afterwards, a lot of mathematicians like

Clebsch, Richmond and Campbell began to study this kind of problems, but

only considering special cases in small degree. Meanwhile, in Italy, the problem

was faced, more in general, by the school of Geometry of Corrado Segre. The

most important results are due to Palatini and Terracini.

I

The work of Terracini is a key point of the story of secant varieties. He

introduced completely new techniques to attack the problem of defectiveness

of projective varieties. The relevance of his work can be understood by the fact

that the famous Terracini’s Lemma is still today the first tool to study secant

varieties. Palatini stated for the first time a conjecture about the defectiveness

of secant varieties of Veronese varieties. He actually knew the entire list of

defective cases, but to have a proof that they are the only cases, we had to

wait almost a century.

In 1985 A. Hirschowitz introduced new and powerful methods. In a joint

work with J. Alexander, they proved the Alexander-Hirschowitz Theorem (1995),

that completely solves the question about the defectiveness of secant varieties

of Veronese varieties, see Theorem 2.9.

In the last decades, the study of defectiveness of some special families of

projective varieties and, more in general, on their secant varieties has received

renew interest. In particular the cases of Segre varieties and Segre-Veronese

varieties.

The Segre product Seg(Pn1C × . . . × PntC ) parametrizes the decomposable

tensors in Cn1+1⊗ . . .⊗Cnt+1, so that, like the polynomial case, the knowledge

of secant varieties of Segre products is strictly connected to the problem of

tensor decompostion, a sort of Waring problem for tensors. Questions about

tensor decomposition are involved in a lot of disciplines including Algebraic

Complexity Theory, Algebraic Statistics, Algebraic Coding Theory or Signal

Processing.

Unfortunately, for Segre products we have not a theorem similar to the

Alexander-Hirschowitz Theorem. The defectiveness of Segre varieties is a very

interesting problem largely unknown. A part for the case of product of two

factors, which is completely known and understood also because it can be

rephrased in terms of matrices, relevant results in the case of Segre product with

many factors are due to M.V. Catalisano, A.V. Geramita and A. Gimigliano.

They have completely solved the case of product of copies of P1, see Theorem

III

3.7, and some specific case under particular numerical assumption, the so-

called “unbalanced” cases, see Theorem 3.9. Moreover, we want to mention H.

Abo, G. Ottaviani and C. Petersen since they classified the defective sth-secant

varieties of Segre varieties for s ≤ 6, see Theorem 3.8.

However, a lot of mathematicians have studied and are currently interested

in this kind of problems. The study of defective varieties could be a very fruit-

ful field of research. Beyond the classical approach, there is a study in terms

of concepts like weakly defectiveness or Grassmann defectiveness in order to

describe the nature of defective varieties. Moreover, there are other projective

varieties related to algebraic objects for what we don’t know almost anything

about the defectiveness or, more in general, about secant varieties. For ex-

ample, Grassmannians, Chow varieties or varieties of spinors. In this lack of

knowledge, it is clear that there is a lot of future work to do.

The aim of this thesis is to give an introduction about the defectiveness

of projective varieties, explaining some methods used to attack this kind of

problems. In Chapter 1 we introduce some terminology that we use in the

core of the thesis. In section 1.4 we define secant varieties and defectiveness

explaining some basic properties about this projective varieties.

In Chapter 2, we deal with secant varieties of Veronese varieties and the

Alexander-Hirschowitz Theorem. Firstly we explain how to reduce the prob-

lem dealing with defectiveness of Veronese varieties to the computation of the

Hilbert function of a set of double points. We use both the classical approach

via Terracini’s Lemma, see Lemma 2.11, and the approach via inverse system

and apolarity, see Theorem 2.5. After that we prove the Alexander-Hirschowitz

Theorem in case n = 2, see Theorem 2.15, and we explain an inductive method

which allows us to prove the theorem in almost all cases, see Theorem 2.18.

The Section 2.3 is devoted to give an idea about the method introduced by

Alexander and Hirschowitz, the so-called “methode d’Horace differentielle”.

In Chapter 3, we consider the case of Segre varieties. In Section 3.2 we use

the Terracini’s Lemma to reduce the problem of defectiveness of Segre variety

to the computation of multigraded Hilbert function of a set of double points in

Pn1 × . . .Pnt . In Section 3.3 we explain a combinatorial approach which gives

results about the dimension of some secant varieties of Segre products. In

Section 3.4 we explain an extremely useful method introduced by Catalisano,

Geramita and Gimigliano which allow us to reduce the problem of computation

of multigraded Hilbert function of set of double points to the computation

of Hilbert function of a scheme in Pn1+...+nt , and so in standard gradation.

In Section 3.6 we introduce the concept of Grassmann secant varieties and

Grassmann defectiveness. We explain the relation between Grassmann secant

varieties and the classical definition of secant varieties and how this relation

can be used to find some result about the defectiveness of projective varieties.

Contents

1 Preliminaries 1

1.1 Graded ring and Hilbert function . . . . . . . . . . . . . . . . . 1

1.2 Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Scheme Theory: an introduction . . . . . . . . . . . . . . . . . . 12

1.3.1 The Zariski topology . . . . . . . . . . . . . . . . . . . . 13

1.3.2 Sheaf Theory: basic definitions . . . . . . . . . . . . . . 15

1.3.3 The structure sheaf of a scheme . . . . . . . . . . . . . . 20

1.3.4 Subschemes . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.8 Projective schemes . . . . . . . . . . . . . . . . . . . . . 33

1.3.10 Invertible sheaves on projective schemes . . . . . . . . . 37

1.3.11 Affine and Projective Tangent Space . . . . . . . . . . . 38

1.4 Secant Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2 Veronese varieties 49

2.1 Hilbert function of fat points via apolarity . . . . . . . . . . . . 51

2.2 Terracini’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.3 Alexander-Hirschowitz Theorem . . . . . . . . . . . . . . . . . . 69

2.3.1 Terracini’s Second Lemma . . . . . . . . . . . . . . . . . 71

2.3.2 The exceptional cases . . . . . . . . . . . . . . . . . . . . 77

2.3.3 Terracini’s inductive method . . . . . . . . . . . . . . . . 80

2.3.4 ”La methode d’Horace differentielle” . . . . . . . . . . . 87

2.4 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

V

CONTENTS CONTENTS

3 Segre varieties 97

3.1 Tensor decomposition and “tensor rank” . . . . . . . . . . . . . 97

3.2 Yet again: Terracini’s Lemma . . . . . . . . . . . . . . . . . . . 101

3.3 A combinatorial approach: the monomial case. . . . . . . . . . . 105

3.4 Multiprojective-affine-projective method . . . . . . . . . . . . . 108

3.5 Higher secant varieties of Segre varieties . . . . . . . . . . . . . 113

3.6 Grassmann secant varieties . . . . . . . . . . . . . . . . . . . . . 114

3.6.1 The map Φ . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.6.2 Dimension of some secant varieties . . . . . . . . . . . . 119

Chapter 1

Preliminaries

The purpose of this chapter is to give the basic notions about the algebraic

and geometric tools that we are going to use in the core of this thesis. We will

give our definitions in the setting that we actually need, without considering

the most general framework.

1.1 Graded ring and Hilbert function

Let R be a commutative ring. We define a Z-graded structure on R by consid-

ering a collection of subgroups Rn ⊂ R for each n ∈ Z such that R =⊕

n∈ZRn

and with a compatibility law with respect to the product, i.e.

Rn ·Rm ⊂ Rn+m.

This view of R as a direct sum of its subgroups means that each element

f ∈ R can be written as a sum

f =∑n∈Z

fn,

where {fn ∈ Rn} is a family with only a finite number of nonzero elements.

Such fn are called homogeneous components of degree n of f .

Remark 1.1. By the compatibility with respect to the product, we can see

that R0 is a ring and, similarly, each Rn is an R0-module. Usually, we will

1

1.1 Graded ring and Hilbert function 1. Preliminaries

work in assumption where R0 will be a field and then each Rn will be a k-vector

space. In general, we may substitute Z with a generic semigroup H. In fact, it

is the unique structure of Z that we have used to define a Z-graded structure.

Let R =⊕

n∈ZRn be a Z-graded ring and I ⊂ R be an ideal. We say that I

is a homogeneous ideal if it satisfies one of the following equivalent conditions:

� I is generated by homogeneous elements;

� for each f =∑

n∈Z fn, we get that

f ∈ I ⇐⇒ fn ∈ I ∀n ∈ Z.

Now, if I is an homogeneous ideal, we can define the subgroup of Rn,

In = I ∩Rn, and hence

R/I =⊕n∈Z

Rn/In.

Remark 1.2. Let R be a Z-graded ring and M be a R-module. We say that M

is an Z-graded R−module if, for each n ∈ Z, there exists an abelian subgroup

Mn such that M =⊕

n∈ZMn with a compatibility to the graded structure of

R, i.e.

Rn ·Mm ⊂Mn+m.

Similarly to the ring case, by this compatibility, each Mn has the structure of

an R0-module.

In order to get more confidence with these definitions, we give some example

of graded structure on the polynomial ring:

Example 1.1. (1) Let k be a field and R = k[x1, . . . , xn] be the polynomial

ring. R is a N-graded ring by defining R0 = k and

Rs = k[x1, . . . , xn]s = 〈all monomials of degree s〉.

This is called the standard graded structure on the polynomial ring.

1. Preliminaries 3

(2) We can define the N-graded structure on the polynomial ring with respect

a vector w = (w1, . . . , wn) ∈ Nn by defining deg(xi) = wi, hence we get

Rs = 〈xα := xα11 · · ·xαnn such that α1w1 + . . .+ αnwn = s〉

and R0 = k [xi | wi = 0] .

(3) Another way to define a different graded structure on the polynomial ring

R is by setting

deg(xi) = ei = (0 . . . 0, 1, 0 . . . 0) ∈ Nn,

hence we get that

deg(xα) = α ∈ Nn.

With notations as above, we have the Nn-graded structure

R =⊕α∈Nn

Aα,

where Aα = 〈xα〉. This is the standard multi-graded structure on R.

(4) More in general, we may consider a matrix W = (wi,j) ∈ Mmn(N) and

define

deg(xi) = wi = (wi,1, . . . , wi,m) ∈ Nm,

hence we get

deg(xα) = α1w1 + . . .+ αnwn ∈ Nm.

In this way we have a Nm-graded structure

R =⊕s∈Nm

As,

where As = 〈 all monomials xα such that deg(xα) = s 〉.

Furthermore, we would like to work with morphisms that respect the graded

structure. Let M1 and M2 be two graded R-modules and let f : M1 → M2 be


a morphism of R-modules. We say that f is homogeneous, or graded, of degree

t if we get

f ([M1]s) ⊂ [M2]s+t.

The most natural graded morphisms are those of degree 0. For this reason we

define an operation that allows to work only with these morphisms.

Let M =⊕

n∈ZMn be a Z-graded ring and consider a ∈ Z, we define the

shifted module M(a) as the Z-graded ring

M(a) :=⊕n∈Z

Mn+a.

In this way, let φ : M → N be a morphism of graded modules of degree d, then

we can shift the module M and get a morphism of degree 0

φ : M(−d) // N.

Using the definition of shifted modules we can introduce a graded version

of the Hilbert Syzygies Theorem. Let M be a graded finitely generated A-

module with generators {m1, . . . ,ms} of degrees {d1, . . . , ds}, respectively. We

can define a graded surjective homomorphism of R-modules

φ : R(−d1)⊕ . . .⊕R(−ds) // M,

by sending the generators ei of Rs to the generators mi of M .

Definition 1.1. Let M a graded R-module, a graded resolution of M is an

exact sequence of the type

. . . // F2// F1

// F0// M // 0,

where, Fi is a graded free R-module ' R(−di,1) ⊕ . . . ⊕ R(−di,si), for any i,

and each map is a graded morphism of degree 0.

Theorem 1.1 (Hilbert graded syzygies Theorem). Let M be a graded

R-module finitely generated with R = k[x1, . . . , xn], then there exists a finite

graded free resolution of M of length, at most, equal to n.

1. Preliminaries 5

1.1.1. Hilbert function. A useful tool to study graded ring and graded

module is the Hilbert function. Since we will always be in this assumption,

we consider a field k, the polynomial ring R = k[x1, . . . , xn] with the standard

gradation and a finitely generated graded R-module M .

Remark 1.3. We just say that, in general, we can talk about the Hilbert func-

tion if have a Z-graded ring R with the piece R0 Artinian, R is Noetherian, so

that it is finitely generated as R0-algebra, and M is a graded finitely generated

R-module.

Coming back to our assumption, we can see that each piece Mi is a k-vector

space with finite dimension. Let {m1, . . . ,ms} be the generators of M as an

R-module with deg(mi) = ti, for i = 1, . . . , s. We get that

Mi =

{s∑i=1

aimi

∣∣∣ ai ∈ Pi−ti},

and then, as k-vector space, it is generated by the finite set of elements

Mi =⟨xd11 · · ·xdnn mj

∣∣∣ d1 + . . .+ dn + tj = i, j = 1, . . . , s⟩.

Now, it makes sense to define the Hilbert function of M as the map

HFM : Z → N

i 7→ dimk(Mi)

Example 1.2. If we consider R as a R-module, it is clear by the definition of

the standard gradation that we get

HFR(i) =

(i+ n− 1

n− 1

).

Lemma 1.2. Let M a graded R-module finitely generated, then

HFM(t) = 0, for all integers sufficiently small.


Proof. By Hilbert Syzygies Theorem, consider a free graded resolution of M

0 // Fr // Fr−1// . . . // F1

// F0// M // 0

Since the morphisms are of degree 0, we can consider the first segment of

the sequence restricted to a fixed grade t

[F0]tφ0 // Mt

// 0.

Since F0 is a free module, we get the natural isomorphism

[F0]t ' [R(−d1)]t ⊕ . . .⊕ [R(−ds)]t,

where the integers d1, . . . , ds are the degrees of each generators mi of M as

R-module, respectively.

By the surjectivity of φ0 we deduce

dim(Mt) ≤ dim[P (−d1)]t + . . .+ dim[P (−ds)]t,

hence, since dim[P (−di)]t = 0 if t− di < 0, we get

dim(Mt) = 0, for all t < mini=1,...,s{di}.

Now, it also makes sense to define the Hilbert series of M :

HSM(z) =∑i∈Z

HFM(i)zi ∈ Q[[z]]z.

Example 1.3. Let M(a) be a shifted module of M , it is obvious that

HFM(a)(i) = HFM(i+ a),

and consequently

HSM(a) = z−aHSM(z).

1. Preliminaries 7

Lemma 1.3. Consider an exact sequence of graded R-modules finitely gener-

ated with each map a graded homomorphism of degree 0

0 // Mt// . . . // M2

// M1// M0

// 0,

then we have

t∑i=0

(−1)iHFMi(j) = 0, for each grade j and

t∑i=0

(−1)iHSMi(z) = 0.

Proof. Since the maps are homomorphisms of degree 0, such exact sequence

induces, in each degree, an exact sequence of k-vector spaces

0 // [Mt]j // . . . // [M2]j // [M1]j // [M0]j // 0,

and than the first equality immediately follows.

For the second one, we can write

t∑i=0

(−1)iHFMi(j)zj = 0 for each degree j ⇒

∑j

t∑i=0

(−1)iHFMi(j)zj = 0.

Since we are working with formal series, we have no problem to switch the

order of series and we get

t∑i=0

(−1)iHSMi(z) = 0.

Example 1.4. To compute the Hilbert function of a quotient of the polyno-

mials ring R/I, the usual case in Algebraic Geometry, we can consider the

resolution of the quotient

0 // I // R // R/I // 0.

In this way, it suffices to compute HFI and than to use the formula

HFR/I(t) = HFR(t)− HFI(t).

1.2 Multilinear Algebra 1. Preliminaries

1.2 Multilinear Algebra

Let k be a field and V be a vector space over k. The purpose of this section

is to recall some basic fact of Multilinear Algebra. In particular, we give the

construction of the tensor algebra and the symmetric algebra associated to V .

1.2.1. Tensor product and tensor algebra. Let V1, . . . , Vr,W be k-vector

spaces.

Definition 1.2. An application f : V1 × . . .× Vr → W is called multilinear if

it is linear in each component, i.e. for each i ∈ {1, . . . , r}

f(x1, . . . , axi + byi, . . . , xr) = af(x1, . . . , xi, . . . , xr) + bf(x1, . . . , yi, . . . , xr).

We define the tensor product V1 ⊗ . . . ⊗ Vr as the k-vector space which is

the unique solution of the following universal problem.

Proposition 1.4. Using the previous notation, there exists a unique couple

(T, g), where T is a k-vector space and g : V1 × . . . × Vr → T is a multilinear

map such that for any multilinear map f : V1 × . . . × Vr → W , there exists a

unique morphism of k-vector spaces f ′ : T → W such that f = f ′ ◦ g, i.e. the

following diagram is commutative:

f : V1 × . . .× Vr //

g

��

W

Tf ′

77oooooooooooooo

Proof. [Uniqueness] Suppose that both of the couples (T, g) and (T ′, g′) solve

the universal problem. Using two times such property we get the diagram

f : V1 × . . .× Vr //

��

T ′

wwoooooooooooooo

T

77oooooooooooooo

and then an isomorphism between T and T ′.

1. Preliminaries 9

[Existence] Let k(V1×...×Vr) be the free k−vector space{n∑i=1

ai

(x

(i)1 , . . . , x

(i)r

) ∣∣∣ n ∈ N, ai ∈ k, x(i)j ∈ Vj, i = 1, . . . , n, j = 1, . . . , r

}.

Let D be the submodule of k(V1×...×Vr) generated by the following kind of ele-

ments

� (. . . , xi−1, xi+yi, xi+1, . . .)−(. . . , xi−1, xi, xi+1, . . .)−(. . . , xi−1, yi, xi+1, . . .);

� (. . . , xi−1, axi, xi+1, . . .)− a(. . . , xi−1, xi, xi+1, . . .);

and set T = k(V1×...×Vr)/D. For each element (x1, . . . , xr) ∈ V1 × . . . × Vr in

the basis, we denote x1 ⊗ . . . ⊗ xr the image by the canonical projection on

the quotient. Thus, T is generated by such elements x1 ⊗ . . .⊗ xr and, by the

definition of D, we get

�

(. . .⊗xi−1 ⊗ xi + yi ⊗ xi+1 ⊗ . . .) =

(. . .⊗ xi−1 ⊗ xi ⊗ xi+1 ⊗ . . .) + (. . .⊗ xi−1 ⊗ yi ⊗ xi+1 ⊗ . . .);

� (. . .⊗ xi−1 ⊗ axi ⊗ xi+1 ⊗ . . .) = a(. . .⊗ xi−1 ⊗ xi ⊗ xi+1 ⊗ . . .);

equivalently, the following application

g : V1 × . . .× Vr −→ T

(x1, . . . , xr) 7−→ x1 ⊗ . . .⊗ xr

is multilinear.

Finally, we can extend a given multilinear map f : V1 × . . .× Vr → W , by

linearity, to a morphism of k-vector spaces f : k(V1×...×Vr) → W . Moreover, f

vanishes on all the generators of D, thus the morphism

f ′ : T −→ P

x1 ⊗ . . .⊗ xr 7−→ f(x1, . . . , xr).

is well-defined and uniquely determined by f . So, we get that the couple (T, g)

solves the universal problem.

1.2 Multilinear Algebra 1. Preliminaries

With this proposition, we can define the tensor product of k-vector space

V1⊗ . . .⊗Vr which is the k-vector space generated by the products x1⊗ . . .⊗xr,

with xi ∈ Vi. Moreover, if each k-vector space Vi is finitely generated, then also

V1 ⊗ . . .⊗ Vr is finitely generated by all the tensor products of the generators.

Remark 1.4. With these definitions and from the universal property, it is an

easy exercise to prove the following useful properties of the tensor product:

� (V ⊗ U)⊗W ' V ⊗ (U ⊗W );

� k ⊗ V ' V ;

�

(⊕i∈I Vi

)⊗ U '

⊕i∈I (Vi ⊗ U) ;

where k is a field and V, U, W, Vi are k-vector spaces.

Now, let V a k-vector space. For each integer i ≥ 0, we define

T s(V ) :=s⊗i=1

V and T 0(V ) = k.

By the associativity of the tensor product, we obtain a bilinear map

T r(V )× T s(V ) // T r+s(V ).

Consequently, we can define a ring structure on the direct sum

T (V ) :=∞⊕i=0

T i(V ),

and then an algebra structure. We call T (V ) the tensor algebra of V. In general,

it is not commutative and we denote with ⊗ the ring operation in T (V ).

1.2.2. Symmetric Algebra. In T r(V ), let Ir be the submodule generated

by all elements of type

x1 ⊗ . . .⊗ xr − xσ(1) ⊗ . . .⊗ xσ(r),

1. Preliminaries 11

for all xi ∈ V and σ ∈ Sr1. We define the quotient

Sr(V ) := T r(V )/Ir,

and let

S(V ) :=∞⊕i=0

S i(V ).

It is immediate to see that

I =∞⊕i=0

Ii

is an ideal in T (V ), and hence S(V ) is a graded k-algebra, which is called the

symmetric algebra of V.

Remark 1.5. An r-multilinear map f : V × . . . × V → W is said to be

symmetric if

f(x1, . . . , xr) = f(xσ(1), . . . , xσ(r)), for all σ ∈ Sr.

In this way, considering the canonical map obtained by composition

V × . . .× V → T r(V )→ T r(V )/Ir = Sr(V ),

we can see the symmetric algebra of V as the solution of the universal problem

for multilinear symmetric map, likewise to the tensor product is the solution

of the universal property for multilinear maps.

The importance of the symmetric algebra in Algebraic Geometry follow

from the next fact.

Proposition 1.5. Let V be a k-vector space of finite dimension and {v1, . . . , vn}

be a basis of V . Then, S(V ) is isomorphic to the polynomial algebra in n vari-

ables with coefficients in k.

Proof. Let t1, . . . , tn be algebraically independent variables over k, and consider

the polynomial algebra k[t1, . . . , tn]. Let w1, . . . , wr be a set of elements in V

such that

wi =n∑j=1

aijvj, for i = 1, . . . , r.

1Sr is the symmetric group on r letters.

1.3 Scheme Theory: an introduction 1. Preliminaries

Then, we define the map

V × . . .× V︸︷︷︸r times

// k[t1, . . . , tn]r

by sending

(w1, . . . , wr)� // (a11t1 + . . .+ a1ntn) · · · (ar1t1 + . . .+ arntn).

Such map is multilinear and symmetric, hence we get the commutative diagram

V × . . .× V

((PPPPPPPPPPPPP// Sr(V )

xxpppppppppp

k[t1, . . . , tn]r

It is clear that the elements vi1 · . . . · vir ∈ Sr(v) maps onto ti1 · . . . · tir ∈

k[t1, . . . , tn]r for any r-tuple of integers (i1, . . . , ir). Since the monomials M(i)(t)

of degree r are linearly independent over k, it follows that the monomials

M(i)(v) ∈ Sr(V ) are linearly independent over k and then the map Sr →

k[t1, . . . , tn]r is an isomorphism. Now, it easy to check that the multiplication

in S(V ) corresponds to the multiplication in the polynomial algebra, and thus

the isomorphism described as above on each component Ss(V ) induces an

isomorphism of algebras from S(V ) onto k[t1, . . . , tn].

1.3 Scheme Theory: an introduction

The basic correspondence in algebraic geometry is the one-to-one relation

{algebraic varieties} oo //{

finitely generated, nilpotent free ringsover an algebraically closed field k

}.

Scheme theory arises if we do not accept the restrictions on the right-hand

side; in other words, we extend the previous relation to the following

{affine schemes} oo // {commutative rings with identity} .

1. Preliminaries 13

1.3.1 The Zariski topology

Let R be a commutative ring with identity, the affine scheme defined from R

will be called Spec(R), the spectrum of R. We define a point of Spec(R) to be

a prime ideal of R

Spec(R) := {℘ ⊂ R | ℘ is a prime ideal}.

To avoid confusion, sometimes we will write [℘] for the point in Spec(R)

corresponding to the prime ℘.

We can observe that, if X is an ordinary affine variety over an algebraically

closed field k and R is the corresponding coordinate ring, the points of X are

precisely the closed points of Spec R, and the closed points contained in the

closure of a point [℘] ∈ Spec R are exactly the points of X contained in the

subvariety of X determined by ℘.

Each element f ∈ R defines a “function” on Spec(R): if x = [℘] ∈ Spec(R),

we denote with κ(x) the quotient field of R/℘, the residue field of Spec(R) at

℘; hence, we define f(x) ∈ κ(x) as the image of f via the canonical map

R // R/℘ // κ(x).

Remark 1.6. In general, the “function” f has values that vary from point to

point; moreover, in general, f is not determined by its values. For example,

the only prime ideal of R = k[x]/(x2) is (x), thus it induces a function which

is 0 at every point of Spec R, albeit it is non-zero.

We define a regular function on Spec(R) simply an element of R.

Now, we want to give a topology on Spec(R). The target is to make each

regular function f ∈ R as much like a continuous function. Since κ(x) vary

from point to point, the usual definition of continuity makes no sense, but each

residue field has a zero element, thus we can talk about the locus of points


such that f(x) is 0. If f is like a continuous function, we would like that such

locus is a closed subset of Spec(R).

Hence, for any subset S ⊂ R, we define

V (S) :={x ∈ Spec(R) | f(x) = 0 ∀f ∈ S} =

= {[℘] ∈ Spec(R) | ℘ ⊃ S} .

Moreover, it suffices to consider V (I) associated to the ideals of R. From

this definition, using basic properties of prime ideals, we get:

Lemma 1.6. (i) Let I, J be ideals in R, then

V (I) ∪ V (J) = V (IJ);

(ii) let {Ii}i∈A a family of ideals in S, then

V

(∑i∈A

Ii

)=⋂i∈A

V (Ii);

So that, they form the family of the closed sets of a topology on Spec(R).

Moreover, if I is the ideal generated by a set S in R, we have V (S) = V (I).

This topology will be called Zariski topology.

Of course, we define the open sets simply as complements of closed sets.

A particular class of open sets is the family of complements of closed defined

from principal ideals. Let f ∈ R, we define the basic open subset

Xf = Spec(R)− V (f) =

= {[℘] ∈ Spec(R) | ℘ 6⊃ (f)} = Spec(Rf ),

where Rf is the localization of R obtained by adjoining the inverse of f .

The family of basic open sets forms a basis for the Zariski topology, namely,

if U is an open subset of Spec(R)

U = Spec(R)− V (S) = Spec(R)−⋂f∈S

V (f) =⋃f∈S

Spec(Rf ).

1. Preliminaries 15

Moreover, since a product is contained in a prime ideal if and only if a factor

is contained, this is a family closed by finite intersections:

t⋂i=1

Spec(Rfi) = Spec(Rg), where g = f1 · . . . · ft.

In general, Spec(R) with Zariski topology is not Hausdorff. In fact, the

only closed points are those corresponding to the maximal ideals.

If we take R as the coordinate ring of a variety X, the closed points in

Spec(R) correspond to the points of X and the closed points contained in the

closure of a point [℘] ∈ Spec(R) correspond to the points of the variety lying

on the subvariety defined by the prime ideal ℘.

1.3.2 Sheaf Theory: basic definitions

Definition 1.3. Let X be a topological space. A presheaf F on X provides

for each open subset U ⊂ X a set, denoted with F (U) or Γ(U,F ), and for

each pair of open subset, V ⊂ U ⊂ X, a restriction map

rUV : F (U)→ F (V )

satisfying the properties:

(i) rUU is the identity map;

(ii) for any triple of open sets U ⊃ V ⊃ W

rVW ◦ rUV = rUW .

The elements of F (U) are called the sections of F (U) and the elements of

F (X) are called global sections.

Remark 1.7. Another way to see this definition is by defining a presheaf F

as a controvariant functor from the category of open sets of X, Top(X), to the

category of sets, Set. Changing the target category, we have the definitions of

presheaf of abelian groups, rings, algebras,...


The definition of presheaf of modules F over a presheaf of rings O on X

is one of the most useful construction for algebraic geometry. Precisely, we

consider

for each open set U ⊂ X a ring O(U) and a O(U)-module F (U),

and for each pair of open set U ⊃ V , a ring of homomorphism of O(U) →

O(V ) and a map of O(U)-modules F (U) → F (V ), by thinking F (V )

as a O(U)-module.

Definition 1.4. A presheaf F is said to be a sheaf if it satisfies the following

“sheaf axiom”:

for each open covering U = {Uα | α ∈ A} of an open set U ⊂ X and

for each collection of elements fα ∈ F (Uα) such that

rUαUα∩Uβfα = rUβUα∩Uβfβ, ∀α, β ∈ A,

there exists a unique element f ∈ F (U) whose restrictions are

rUUαf = fα, ∀α ∈ A.

If F is a presheaf (respectively a sheaf) on X and U is an open subset in X,

we can define the prefisheaf (respectively the sheaf) on U called the restriction

of F to U , by setting

F∣∣U

(V ) = rUV (F (U)) = F (V ), for all opens subset V ⊂ U.

From now, we will work only with sheaf and presheaf with an algebraic

structure, at least of abelian groups, and then it makes sense to define the

direct sum and the tensor product :

F ⊕ G (U) := F (U)⊕ G (U);

F ⊗ G (U) := F (U)⊗ G (U).

Remark 1.8. The direct sum preserves the sheaf axiom, but the tensor prod-

uct may be not a sheaf even if both F and G are sheaves.

1. Preliminaries 17

Example 1.5. The simplest sheaf on a topological space X is the sheaf of

locally constant functions with values in a set S; if S is a group (respectively a

ring), we can define the pointwise sum (and the pointwise product) and makes

it a sheaf of abelian groups (respectively of rings).

When on the set S there is a topology, we can also define the sheaf of

continuous functions on X with values in S, usually denoted with C.

Another way to describe a sheaf is by “stalks”:

Definition 1.5. let F be a presheaf on X and let x ∈ X, we define the stalk

Fx of F on x to be the direct limit

Fx : = lim−→ x∈UU⊂X

F (U) =

=

the disjoint union of F (U) over all open sets U containing x

modulo the equivalence relationσ∼τ if σ∈F (U), τ∈F (V ) and there exists an open set

W⊂U∩V such that x∈W and rUWσ=rVW τ

In this way, for every x ∈ U there is a map

F (U) // Fx

by sending a section s ∈ F (U) to the class of equivalence of (U, s) = sx = [s].

Moreover, if F is a sheaf, a section s ∈ F (U) is determined by its image in

the stalks Fx for each x ∈ U . In fact, by the sheaf axiom

s = 0 if and only if sx = 0, ∀x ∈ U.

Definition 1.6. Let F and G be sheaves on X. A morphism of sheaves

φ : F → G is defined to be a collection of maps φ(U) : F (U) → G (U) for

each open set U ⊂ X, such that, for any pair V ⊂ U the following diagram

commutes:

F (U)φ(U) //

rUV��

G (U)

rUV��

F (V )φ(V ) // G (V )


A morphism of sheaves induces a map of stalks for each x ∈ X

φx : Fx// Gx,

moreover, using the sheaf axiom, the morphism of sheaves is determined by

the image of the induced maps φx.

Definition 1.7. We say that a morphism of sheaves φ : F → G is injective

(resp. surjective and bijective) if the induced maps φx are injectives (resp.

surjectives and bijectives) for each x ∈ X.

Remark 1.9. One can think that such properties can be viewed also at the

level of the morphisms φ(U) for each open U ⊂ X. We don’t investigate more

deeply this fact, but we could proved that:

· φ is injective if and only if all the φ(U) are injectives;

· φ(U) can be not surjective even if φ is surjective.

Definition 1.8. Let F be a presheaf, we define the sheafification F of F to

be the unique sheaf with a morphism of preasheaves φ : F → F such that for

all x ∈ X, the induced map φx is an isomorphism.

More explicitly, for each section σ ∈ F (U) there exists an open cover of U ,

U =⋃α Uα, and sections sα ∈ F (Uα) such that

σx = (sα)x, ∀x ∈ Uα.

Definition 1.9. We will say that F is a subsheaf of G if there exists an

injective map of sheaves φ : F → G .;

If F and G are presheaves and F injects in G , we may define the quotient

as the presheaf defined on each open set U ⊂ X as

G /F (U) := G (U)/F (U),

but if F and G are sheaves, such quotient is not necessarily a shef, hence we

define the quotient between two sheaves as the sheafification of G /F .

1. Preliminaries 19

In our applications to schemes, we will often have a given basis B for the

open sets of X and we want to define a sheaf F just by setting the F (U) for

any U ∈ B and rUV just between U, V ∈ B.

We say that F is a B-sheaf if it satisfies the axioms of sheaves with respect

to the open sets of B.

Proposition 1.7. Let B be a basis for the open sets in X, then

(i) every B-sheaf can be extended to a sheaf on X;

(ii) let F and G be sheaves on X and

φ(U) : F (U)→ G (U), for all U ∈ B,

commuting with restrictions, then there is a unique morphism φ : F → G

of sheaves which extends φ.

Sketch of the proof. The definition of the sheaf F on X is given, for any open

set U ⊂ X, as the inverse limit of F (V ) with V a open set in B which contains

U :

F (V ) : = lim←−V⊂UV ∈BF (V ) =

=

(fV ) ∈∏V⊂UV ∈B

F (V )∣∣∣ rVW (fV ) = fW whenever W ⊂ V ⊂ U, and W,V ∈ B

.

Corollary 1.8. Let U be an open covering of X and FU be a sheaf on any

U ∈ U . Let also

φU,V : FU

∣∣U∩V → FV

∣∣U∩V

be isomorphisms for any pair U, V ∈ U such that

φV,W ◦ φU,V = φU,W , for all triple W ⊂ V ⊂ U in U .

Then, there is a unique sheaf F on X which extends (FU)U∈U .

In this way, to define a sheaf on a topological space (like a scheme), we can

just express the sheaf on a open cover of X.


1.3.3 The structure sheaf of a scheme

Let us coming back to affine schemes. Consider X = Spec(R) with R a com-

mutative ring with unit.

Let Xf be the basis open set defined previously. We define

OX(Xf ) := Rf

and if Xf ⊃ Xg, namely there exists a power of g which is multiple of f , we

define the restriction map:

rXfXg

: Rf// Rfg = Rg.

From the general theory of sheaves, to define the structure sheaf OX , it suffices

to prove that it satisfies the sheaf axioms with respect to the basic open sets

of X.

Proposition 1.9. Let X = Spec(R) and suppose that Xf is covered by

(Xfα)α∈A ⊂ Xf , then

(i) if g, h ∈ Rf and g ≡ h in each Rfα , then they are equal;

(ii) if {gα ∈ Rfα | α ∈ A}, such that for any pair α, β ∈ A, gα ≡ gβ in Rfαfβ ,

then there exists an element g ∈ Rf whose restrictions at Rfα are gα for

any α ∈ A.

In this way, if B is the family of basic open sets of X, OX is a B-sheaf and

it can be extended to a sheaf on X. The elements of OX(U) for some open set

U ⊂ X are called rational functions on X.

More in general, we can define a scheme as follows.

Definition 1.10. A scheme is a topological space X, called the support of the

scheme and often indicated with |X|, with a sheaf of rings OX , such that the

pair (|X|,OX) is “locally affine”, namely there exists a open cover U = {Ui}

of X and a family of rings {Ri} such that

Ui ' |Spec(Ri)| and OX

∣∣Ui' OSpec(Ri).

1. Preliminaries 21

A regular function on an open set U ⊂ X is a section of OX(U) and a global

regular function is a global section of OX(X).

The stalks OX,x of the structure sheaf are called local rings of OX and the

residue field of OX,x is denoted with κ(x).

1.3.4 Subschemes

Let U ⊂ X be an open subset of a scheme X, the pair(U,OX

∣∣U

)is again a

scheme. First consider U = Xf a basic open set, then

(U,OX

∣∣U

)= (Spec(Rf ), Rf ),

so that it is an affine scheme. In general, we recall that the basic open sets

contained in U cover U , so that(U,OX

∣∣U

)is covered by affine schemes and

hence it is a scheme.

We define an open subscheme of X as an open set of the support |X| with

the induced structure.

The definition of closed subsets is more subtle. Let X = Spec(R) an affine

scheme. If I ⊂ R is an ideal, we may think V (I) ⊂ X as an affine scheme by

considering Y = Spec(R/I). Since the primes of R/I are exactly the primes of

R which contains I modulo I, we have that V (I) and Y are homeomorphic.

We define a closed subscheme of X to be a spectrum of a quotient ring of

R, so that we have a correspondence one to one between closed subschemes in

X and ideals in R.

After this definition, we are able to introduce the standard operations be-

tween closed subset. Let X = Spec(R/I) and Y = Spec(R/J):

· X contains Y if Y , in turn, is a closed subscheme of X, in other words if

V (J) ⊂ V (I);

· the union

X ∪ Y = Spec (R/I ∩ J) ;


· the intersection

X ∩ Y = Spec (R/I + J) ;

Remark 1.10. To extend these definitions to an arbitrary scheme, the first

step is to replace the ideal I with a sheaf of ideal I . In the case of an affine

scheme X = Spec(R), the ideal sheaf of Y = Spec(R/I) is defined on each

basic open set Xf as

I (Xf ) = IRf ,

and we can identify the structure sheaf of Y as the quotient

OY = OX/I .

Now, if X is an arbitrary scheme, a closed subscheme Y of X is defined as a

closed topological subset of the support |X| together with a sheaf of rings OY

that is a quotient of OX by a “quasi-coherent” ideal sheaf such that Y ∩ U is

the closed subscheme associated to I (U). The hypothesis of “quasi-coherence”

is quite technical and arises from the general sheaves theory: a sheaf of OX-

modules is called quasi-coherent if, for any open set U and any open basic set

Uf such that Uf ⊂ U , the OX(Uf )-module F (Uf ) is obtained from F (U) just

by adjoining the inverse of f .

One of the most important closed subscheme of an affine scheme is the

reduced scheme associated to X defined as

Xred := Spec(Rred),

where Rred = R/nil(R) and nil(R) is the nilradical ideal of R, i.e. the ideal of

the elements in R which have a null power, usually called nilpotent elements.

Recall that nil(R) =⋂℘∈SpecR ℘ and, therefore, topologically speaking, |X|

and |Xred| are identical. We say that a scheme is reduced if X = Xred.

Remark 1.11. To generalize this definition, let N ⊂ OX the ideal sheaf whose

values on each open U ⊂ X is exactly the nilradical of OX(U):

N (U) := nil(OX(U)).

1. Preliminaries 23

Since the construction of the nilradical commutes with localization, N is a

quasi-coherent ideal sheaf and then it defines a closed subscheme of X.

Definition 1.11. A scheme X is irreducible if |X| is irreducible as topological

space.

Since the irreducibility for topological space is equivalent to say that every

open sets are dense, we get that a scheme Spec(R) is irreducible if and only

if R has a unique minimal prime, or equivalently if the nilradical is prime.

Moreover, Spec(R) is irreducible and reduced if and only if R is a domain.

1.3.5. The local ring at a point.

Definition 1.12. We say that a scheme X is Noetherian if it has a covering

by open affine subschemes which are the spectra of Noetherian rings.

The local ring of a scheme X in x is

OX,x := lim−→x∈UOX(U),

the maximal ideal mX,x is the set of all sections that vanish at x.

Since we can consider an affine open neighborhood of x, we may assume

R = Spec(R) and x = [℘]. We consider the basic open set Spec(Rf ) such that

f /∈ ℘ and we have

OX,x := lim−→f /∈℘Rf := R℘

mX,x := lim−→f /∈mRf := Rm

This definition of the local ring of a scheme at a point is crucial to introduce

some geometrical notions. Let X be a scheme, we define dimension of X at a

point x :

dimxX := Krull dimension of OX,x,

and then we define the dimension of X as the supremum of these local dimen-

sions.


Example 1.6. (i) If a scheme is 0-dimensional, then it is a discrete set.

Indeed, suppose X = Spec(R) and x = [℘] ∈ R, if OX,x has Krull dimen-

sion 0, it means that R℘ has only one maximal ideal ℘R℘, or equivalently,

in R does not exist a prime contained in ℘. From this fact, it means that

we have the following case

Spec(R) = {mi | mi maximal ideal, i ∈ I},

and hence, it has a discrete support. Since a scheme is locally affine, we

get the conclusion in general.

(ii) If a scheme is Noetherian and 0-dimensional, then it is finite.

In fact, let X = Spec(R) an affine scheme, we have already proved that it

is discrete; supposing that it is infinite, it means that the set of maximal

ideals in R is infinite. In this way we can easily get the contradiction:

since it is a Noetherian ring with Krull dimension 0, it is also an Artinian

ring, thus it satisfies the descending chain condition, but

m1 ) m1 ∩m2 ) m1 ∩ . . . ∩m3 ) . . .

should be an infinite descending chain, absurd.

1.3.6 Examples

Now we show some examples in order to begin to understand the “geometry”

related to schemes that we are going to use in the next chapters.

As usual, let k be an algebraically closed field. We define the affine n-space

over k as the scheme

Ank := Spec k[x1, . . . , xn].

Theorem 1.10 (Hilbert Nullstellensatz). Let k be a field. If m is a

maximal ideal in k[x1, . . . , xn] (or equivalently a closed point P of the scheme),

then the quotient k[x1, . . . , xn]/m = κ(P ) is a finite dimensional k-vector space.

1. Preliminaries 25

In our case, since k is algebraically closed, we get

k[x1, . . . , kn]/m = κ(P ) = k,

thus, writing λi for the image of xi in κ(P ), we have that

m = (x1 − λ1, . . . , xn − λn).

In this way the closed points of Ank as scheme correspond to n-tuples of elements

of k.

Let us begin with low dimensions: consider the affine line

A1k = Spec k[x].

It obviously contains a closed point for each value λ ∈ k, which correspond

to the points of the affine line thought as affine variety, but the scheme A1k

contains one more point, the so-called generic point of A1k associated to the

ideal (0). The closure of this point is clearly the entire affine line and hence,

the closed subset of the line A1k are the finite subset of A1

k r {0}.

Let now

A2k = Spec k[x, y], the affine plane.

Similarly to the previous case, it is the standard A2k thought as variety with

some additional points. Again, we have that the closed points (λ, µ) of the

plane come from the maximal ideals (x−λ, y−µ), but we have two more kind

of non-closed points:

· let f(x, y) ∈ k[x, y] be an irreducible polynomial, it corresponds to an irre-

ducible hypersurface in the plane, but in the scheme theory we have a

point corresponding to the prime ideal ℘ = (f) ⊂ k[x, y]. Its closure

consists of the point itself union with all the closed point (λ, µ) such

that f(λ, µ) = 0, in other words, it is the generic point of the variety

associated to the ideal ℘;

· a point corresponding to the ideal (0): the generic point of A2k.


The situation with the general affine space Ank is similar. Geometrically,

it is the classical Ank thought as affine variety with an additional point ℘Σ for

each positive dimensional subvariety Σ ⊂ An. As the previous cases, ℘Σ lies

on the closure of the locus of closed points in Σ and contains all these points,

as well as the generic points of the subvarieties of Σ.

More generally, if X ⊂ Ank is an affine subvariety associated to the ideal

I ⊂ k[x1, . . . , xn] and coordinate ring R = k[x1, . . . , xn]/I, we can associate to

X the affine scheme Spec(R). This scheme is obtained by X by adding one

point ℘Σ for each positive dimensional subvariety Σ ⊂ X.

1.3.7. Non reduced schemes. Now we look at some examples of affine

schemes Spec(R) where R is a finitely generated algebra over an algebraically

closed field, but it may have some nilpotents elements.

The simplest of such scheme is X ⊂ A1k defined by the ideal (x2), that is

X = Spec(k[x]/(x2)

),

so that this scheme has only one point corresponding to the prime ideal (x),

but it is different from the scheme Spec (k[x]/(x)) .

As subschemes of the affine line A1k, we can see this difference by observing

that a function f ∈ k[x] vanishes on X if and only if both of f and ∂f/∂x vanish

at 0. Possibly for this reason, X is sometimes called first-order neighborhood

of 0 in A1k. Usually, we will call X the double point in A1

k with support at the

origin O and often we will denote it with 2O. In general, we will call 2P with

P = (λ) the subscheme of A1k associated to the ideal (x− λ)2.

Moreover, for any n the ideal (xn) defines a subscheme X ⊂ A1k with co-

ordinate ring k[x]/(xn) and a function f ∈ k[x] vanishes at X if and only if f

and the first n− 1 derivatives of f vanish at 0. We will call it the n-fold point

of A1k with support at O and we will denote it with nO.

1. Preliminaries 27

The next step is to consider the idea of a double point of A1k with support

in the origin, like above, but embedded in the affine plane A2k = Spec k[x, y].

Hence, we are looking for a subscheme Y ⊂ A2k such that R = OY (Y ) '

k[ε]/(ε2) is the coordinate ring and φ : k[x, y] → R is the surjection defining

the inclusion of Y in A2k. The inverse image of the unique maximal ideal m of

R is (x, y) ⊂ k[x, y] corresponding to the origin, and since m2 = 0, the map φ

vanishes on (x, y)2 = (x2, xy, y2) and hence factors through a map

φ : k[x, y]/(x2, xy, y2) // R,

or equivalently, the subscheme Y must be contained in

Spec k[x, y]/(x2, xy, y2).

Now, the ring k[x, y]/(x2, xy, y2) is a 3-dimensional k-vector space, whereas

R is only 2-dimensional: it follows that the kernel of φ contains a nonzero

homogeneous linear form αx + βy. So, the scheme that we are looking for

should be of the type

Xα,β = Spec k[x, y]/(x2, xy, y2, αx+ βy) ⊂ A2k.

To understand the geometrical meaning of such scheme we observe that it

can be characterized as follows:

· Xα,β is the subscheme in the affine plane associated to the ideal of functions

that vanish at the origin and satisfy the equation

β∂f

∂x− α∂f

∂y= 0;

· it is the image of the scheme X = Speck[x]/(x2) ⊂ A1k via the inclusion given

by

x 7→ (βx,−αx).

In the classical language, the subscheme Xα,β is said to be the point (0, 0)

together with an arrow pointing the direction defined by the line αx+ βy = 0,


Figure 1.1: The scheme Xα,β in A2k.

or similarly with an “infinitely near point” in such direction. We can draw

such scheme as in Figure 1.1.

At this point, a natural question could be:

how do the schemes of the type Xα,β arise?

When we study the intersection between a conic C and a tangent line L,

we say that “they should meet twice”. We would like that such intersection

determines L, exactly as it happens in the non-tangent case. We know that if

the intersection between L and C are two distinct points, then they determine

uniquely the secant line.

For example, let L : y = 0 and C : y = x2.

The intersection is the scheme

Speck[x, y]/(y − x2, y) = Speck[x, y]/(y, x2).

and this is exactly the scheme X0,1

which we have defined above and the line L is completely determine by the

point X0,1.

Another way to see the scheme Xα,β in A2k is as limit of reduced schemes.

Let

X = {(0, 0), (a, b) 6= (0, 0)} ⊂ A2k,

1. Preliminaries 29

we have that such scheme can be viewed as X = Spec S where

S = k[x, y]/((x, y) ∩ (x− a, y − b)) =

= k[x, y]/(x2 − ax, xy − bx, xy − ay, y2 − by)

Now, suppose (a, b) moves to (0, 0) along a curve (a(t), b(t)) with

(a(0), b(0)) = (0, 0) and a, b ∈ k[t].

We can consider the scheme Xt associ-

ated to the ideal

It = (x, y) ∩ (x− a(t), y − b(t))

and we think the limit for t → 0 of Xt

as the scheme associated to the limit

of It. They are all k-vector spaces of

codimension 2 in k[x, y], hence it

makes sense to compute the limit. Clearly the generators of It have as the

polynomials x2, xy, y2 as their limits, so these are in the limit I. Moreover,

all the ideals It contains the linear form

a(t)y − b(t)x = (xy − b(t)x)− (xy − a(t)y)

and hence, for t 6= 0 the polynomial

a(t)y − b(t)xt

= a1y − b1x+ t(. . .).

In this way the ideal I contains the limit a1y−b1x. Thus, I ⊃ (x2, xy, y2, a1y−

b1x), but since both of them are k-vector spaces of codimension 2, the equality

holds.

In summary, I = (x2, xy, y2, a1y − b1x) and it correspond to the scheme

Xα,β = limt→0

Xt, with α = b1, β = a1,

in other words, it is the origin union the direction of the tangent line of a curve

(a(t), b(t)) in (0, 0).


Since the scheme Xα,β can be viewed as the origin together with a direction,

we will say that it has length or degree 2. We will reserve the name double point

2O in A2k for the scheme with support at the origin, but which are the first-

order neighborhood of O in the sense that a function f vanishes at 2O if and

only if both of f and all its partial derivatives vanish at O.

More in general, if X = Spec R and R is a finite dimensional k-vector space,

we define the degree of X, say deg(X), to be the dimension of such vector space.

Up to now we have considered only schemes of degree 2. Now we want to

talk about schemes of degree 3 or more. Hence, we are going to explain better

what we mean for multiple point in the general affine space Ank .

Remark 1.12. The case of schemes of degree 2 is quite “trivial” in the sense

that

all schemes of degree 2 over an algebraically closed field are isomorphic,

i.e. every local k-algebra of dimension 2 is isomorphic to k[x]/(x2).

Proof. Let m the maximal ideal of R, R/m ' k and, since R is 2-dimensional,

m has dimension 1. By Nakayama’s Lemma m2 = 0, so that the map of k-

algebras k[x] → R by sending x to the unique generator of m has x2 in the

kernel and identifies R with k[x]/(x2).

Now, let Z = Spec k[x1, . . . , xn]/I = Spec R ⊂ Ank be a 0-dimensional

subscheme with support at the origin, namely there is only the maximal ideal

(x1, . . . , xn) and no other primes, of degree 3. If m is the unique maximal ideal

of R, since k is algebraically closed, R/m ' k and then m has dimension 2.

Hence, we could have two cases:

(i) if m is generated by two independent linear forms, we may write m = 〈x, y〉

and consider the map k[x, y]→ R defined by sending the generators x, y

to the generators of m; by Nakayama’s Lemma, m2 = 0, thus the ideal

(x2, xy, y2) is in the kernel of φ and we get the isomorphism

Z ' k[x, y]/(x2, xy, y2);

1. Preliminaries 31

(ii) otherwise, if m is generated by m = 〈x, x2〉, we consider the map k[x]→ R

defined like above; by Nakayama’s Lemma m2 = 0, thus x3 is in the kernel

of such map and we get the isomorphism

Z ' k[x]/(x3).

Moreover, these two cases are not isomorphic because in the first case each

non-zero element has null square, instead in the second case x2 is non-zero in

k[x]/(x3).

Also in this case, both of these kind of non-reduced scheme can be realized

as limits of triples of distinct points. We don’t explicit the proof of this fact, but

the two cases that we have explained above, geometrically, can be expressed in

the figures 1.2 and 1.3.

Figure 1.2: Case (i) Figure 1.3: Case (ii)

In particular, the first case is very interesting: the feature is that the ideal

which defines such scheme is a power of an ideal which defines a simple point.

This is exactly what we mean for multiple point in Ank and this kind of scheme

will be one the most important topics of this thesis (actually, we will work with

the projective analog of these one).

Just to be more precise we give the following definition.


Definition 1.13. Let P ∈ An be a point associated to the ideal ℘ ⊂ k[x1, . . . , xn].

The t-fat point is the non-reduced 0-dimensional subscheme of An defined by

the ideal ℘t ⊂ k[x1, . . . , xn].

The notions t-fat point, t-tuple point or tP will be equivalent.

Remark 1.13. With this definition is clear that a st-fat point tP can be

thought as the tth-order neighborhood of the point P in the sense that a func-

tion f vanish at tP if and only if f and its partial derivatives of order less than

t vanish.

Moreover, we can immediately understand by the definition that a t-fat

point in Ank has degree equal to

(t+n−1n

). In the particular case of a double

point 2P in Ank we can thought about it as the point P together with an arrow

for each direction given by x1, . . . , xn like in the following figures.

Figure 1.4: 2O in A1k. Figure 1.5: 2O in A2

k. Figure 1.6: 2O in A3k.

Definition 1.14. More in general, we will talk about 0-dimensional sub-

schemes in An of fat points, so that we will consider the subschemes associated

to the intersection of powers of ideal of points, namely Z = m1P1+. . .+msPs ∈

An and then

IZ = ℘m11 ∩ . . . ∩ ℘mss ⊂ k[x1, . . . , xn].

1. Preliminaries 33

1.3.8 Projective schemes

Once we have explain the affine schemes, the theory of projective schemes is

quite similar: it differs from the theory of projective varieties in ways that are

analogous to the difference between affine varieties and affine schemes. Like

the affine case we will define the projective spaces and then we will talk about

their subschemes. Let R a commutative ring with identity, we consider an

N-graded finitely generated R-algebra as we have seen in the first section,

S =∞⊕i=0

Si.

Consider the ideal generated by the homogeneous elements of strictly positive

degree

S+ =∞⊕i=1

Si,

usually called the irrelevant ideal. The underlying topological space |Proj S|

is the set of homogeneous prime ideals that not contain S+. The topology is

define like the affine case. The closed sets associated to any homogeneous ideal

I ⊂ S is

V (I) := {℘ ∈ Proj S | ℘ ⊃ I and ℘ 6⊃ S+}.

We will give to |Proj S| the structure of a scheme by specifying the structure

on a basis open sets. Going back over the definitions given in the affine case,

let f be an element of S of degree 1, thus we define the open set

Uf = |Proj S| − V (f)

of homogeneous primes of S not containing f and thus the irrelevant ideal S+.

The points of Uf may be identified with the homogeneous primes of S[f−1].

Moreover, we can observe that if I is a homogeneous ideal in S, since f is of

degree 1, the intersection

IS[f−1] ∩ S[f−1]0

is generated by the elements obtained multiplying a set of generators of I by

appropriate negative powers of f and hence, the homogeneous primes of S[f−1]

are in one-to-one correspondence with the primes of S[f−1]0.


It follows that we can give to Uf the structure of affine scheme by identifying

it with Spec S[f−1]0. We will write (Proj S)f for this open affine subscheme

of Proj S.

Remark 1.14. If x0, x1, . . . are elements of degree 1 and generators of an ideal

I whose radical is the irrelevant S+, the sets

(Proj S)xi = Proj S − V (xi)

forms an affine open cover of Proj S.

Proof. The open sets (Proj S)xi cover Proj S if and only if each prime homo-

geneous ideal doesn’t contain the ideal I. Now, since the radical√I is equal

to S+, if there exists a homogeneous prime ideal ℘ ∈ Proj S wich contains all

the generators xi, we get the inclusion M ⊂ ℘ and, since the radical preserves

inclusions,√M = S+ ⊂

√℘ = ℘, which is a contradiction.

Let f, g ∈ S1, then the intersection (Proj S)f ∩ (Proj S)g is an open affine

subset of (Proj S)f given by

S[f−1]0[(g/f)−1] = S[f−1, g−1]0.

By symmetry in f and g, we get

((Proj S)f )g/f = ((Proj S)g)f/g, (1.1)

and then it makes Proj S into a scheme as we have defined it previously.

Considering the polynomial ring over a field k,

S = k[x0, . . . , xn],

we define Proj S as the projective n-dimensional space which we will indi-

cate with PnR. By definition, we get that the S+ = (x0, . . . , xn), hence a open

covering is formed by the open sets of the type

Ui = Spec S[x−1i ]0 = k[x′0, . . . , x

′n],

1. Preliminaries 35

where x′j = xj/xi. In this way we have x′i = 1 and then, as for the projective

space in the meaning of varieties,

(Pnk)xi = Ank .

1.3.9. Closed subschemes of Pnk . Let I ⊂ S = k[x0, . . . , xn] be an homo-

geneous ideal: for each open of the basis Ui defined above, we can define the

ideal

I(Ui) = IS[x−1i ] ∩ S[x−1

i ]0,

in this way, by the equality (1.1), we get that I satisfies the axiom of sheaf with

respect the open basis {Ui}. By the sheaf theory that we have seen previously,

we get that I is a sheaf and moreover a quasi-coherent sheaf. Consequently,

we can talk about the closed subscheme V (I) of Pnk associated to the ideal I,

we will indicate it also with V+(I).

Now, we can observe that

V+(I) ∩ Proj(S)xi = V (IS[x−1i ] ∩ S[x−1

i ]0) =

= Spec S[x−1i ]/IS[x−1

i ] = Spec S/I[x−1i ] = (Proj S/I)xi ,

in other words,

V+(I) ⊂ Pnk is isomorphic to the scheme Proj S/I.

Conversely, if X is a subscheme of Pnk , we can define the ideal I(X) gener-

ated by the following family of polynomials

I(X) = (P (x0, . . . xn) | ∀ i, P (x′0, . . . , 1, . . . , x′n) ∈ IX(Ui)) ⊂ k[x0, . . . , xn].

Now, we can observe that the ideal sheaf defined from I(X) is exactly the ideal

sheaf IX , in other words

V+(I(X)) = X. (1.2)


Remark 1.15. In the projective case, the correspondence between ideals and

subschemes is not one-to-one: for example, in P1k, the ideal I = (x0) and

J = (x20, x0x1) both define the same subscheme, the reduced point P1 = [0 : 1].

More in general, we can prove that if I is an homogeneous ideal and J =⊕n≥n0

In, then I and J define the same scheme.

Proof. Since J ⊂ I, we get easily the inclusion V+(I) ⊂ V+(J); conversely, let

℘ ⊃ J and suppose that there exist g ∈ I r ℘: since g ∈ I, gn0 ∈ J ⊂ ℘ and,

by the definition of prime ideal, we get the contradiction, g ∈ ℘.

To avoid this inconvenience, we need to introduce the following definition:

Definition 1.15. Let I ⊂ S an homogeneous ideal, the saturation of I is the

ideal

Isat = {F ∈ S | F · Sn ⊂ I, for some positive integer n},

we say that I is saturated if I = Isat.

Remark 1.16. With this definition we get the one-to-one correspondence:

{saturated homogeneous ideals of S = R[x0, . . . , xn]} {subschemes of Pnk}//oo

Proof. Let X be a subscheme of Pnk : the ideal I(X) is a saturated ideal.

In fact, suppose that there exists a positive integer m such that for all

i = 1, . . . , n

P (x0, . . . , xn) · xmi ∈ I(X),

then P (x′0, . . . , 1, . . . , x′n) ∈ IX(Ui) for all i, and hence P ∈ I(X).

Now, let I be an ideal in R[x0, . . . , xn]: I (V+(I)) = Isat.

Since we have the trivial inclusion I ⊂ I (V+(I)) and the passage to the

saturation holds the inclusions,

Isat ⊂ I (V+(I)) ;

1. Preliminaries 37

conversely, let P ∈ I (V+(I)) , by definition we get P (x′0, . . . , 1, . . . , x′n) ∈ I[x−1

i ]

for all i = 1, . . . , n, which means that P (x′0, . . . , 1, . . . , x′n) · xmii ∈ I for some

positive integer mi; consequently, we get that P ∈ Isat, thus

I (V+(I)) = Isat. (1.3)

Using both of the equation (1.2) and (1.3), we get the one-to-one relation

between subschemes and saturated ideal which we were looking for.

1.3.10 Invertible sheaves on projective schemes

We have introduced previously the theory of sheaves in order to define the

concept of scheme. We have also defined the sheaf of the regular function on a

scheme X denoted with OX .

In this section we are going to introduce the general definition of invertible

sheaf which will be useful to define a family of invertible sheaf on a scheme X.

Such kind of sheaves will be fundamental in the following chapters.

Definition 1.16. Let X be a scheme and OX the sheaf of regular functions on

X. A sheaf F of OX-modules is said to be free if it is isomorphic to a direct

sum of copies of OX . It is called locally free if X can be covered with open

sets U for which F (U) is a free OX(U)-module. In that case, the rank of F

in each open U is the number of copies of OX(U) needed. If X is connected,

the rank is the same everywhere.

A locally free OX-module is said to be an invertible sheaf if it has rank 1.

Let now M be an S-graded module on the graded ring S. We define the

associated sheaf to M on Proj S, denoted by M , as follows (cfr. with the

definition of the Proj). For each ℘ ∈ Proj S, we consider the localization of

M℘ at the homogeneous elements which are not in ℘ and then consider the

group of elements of degree 0, [M℘]0. Now, we define M as the set of function

s from U to∐

℘∈U [M℘]0 which are locally fraction, i.e. for every ℘ ∈ U there is


a neighborhood of V of ℘ such that for every ℘′ ∈ V , there exist m ∈ M and

f ∈ S such that we have f /∈ ℘′ and s(℘′) = m/f ∈ [M℘′ ]0. We make M into

a sheaf with the obvious restriction maps.

Definition 1.17. Let now S a graded ring and let X = Proj S. We define the

sheaf OX(d) to be the associated sheaf of OX-modules to Sd. For any sheaf F

of OX-modules, we define the sheaf F (d) as the tensor product F ⊗ OX(d).

Moreover, we may define the graded S-module associated to F to be

Γ∗(F ) =⊕d∈Z

Γ(X,F (d)).

The graded S-module structure is defined as follows. Consider s ∈ Sd. It

determines naturally a global section of OX(d), then for a given t ∈ Γ(X,F (e))

we define the product s · t as the tensor product s ⊗ t and using the natural

map F (e)⊗ OX(d) ' F (d+ e).

Proposition 1.11 (sull’ Hartshorne). If S = A[x0, . . . , xn] and X = Proj S,

then

Γ∗(OX) ' S.

In this way it is natural to think as OPn(d) as the forms in n+ 1 variables

and degree d, i.e. hypersurfaces of degree d in Pn.

1.3.11 Affine and Projective Tangent Space

The basic idea is to give an analogous definition to the corresponding one for

differential geometry. Let X ⊂ An and P ∈ X, a naive definition as the tangent

space to X at P could be the set of all lines through P and tangent to X.

Suppose I(X) = (f1, . . . , fs) the ideal associated to X and P = (0, . . . , 0) ∈

An: if the line is l = {(at) = (a1t, . . . , ant) | t ∈ k}, the condition of tangency

is that fi(at) = tLi(a) + (terms of degree ≥ 2) is divisible by t2, i.e. we have a

double intersection at P , which means that

L1(a) = . . . = Ls(a) = 0.

1. Preliminaries 39

The geometric locus of points on tangent lines to X at P = (p1, . . . , pn) is

the tangent space and it is denoted with TPX; for the previous argument we

get that the equation of the tangent space are

∑j

∂f1

∂xj(P )(xj − pj) = . . . =

∑j

∂fs∂xj

(P )(xj − pj) = 0.

Such linear forms are called the differential of fi at P and are often denoted

with dPfi.

Suppose now g ∈ OX a regular function on X, or the restriction to X

of a polynomial G, we may consider the differential dPG: since for any f ∈

(f1, . . . , fs) the differential dPf is a linear combination of the differentials dPfi

which are zero on the tangent space TPX, we get also dPf = 0 on the tangent

space. Hence, we may define the following linear form on TPX

dPg = dPG∣∣TPX

,

and it does not depend on the polynomial G. In other words, we get an

homomorphism

dP : OX −→ (TPX)∗,

which can be restricted to dP : mP,X → (TPX)∗, since dPα = 0 for all α ∈ k.

With this homomorphism, we can prove that we have the isomorphism

mP,X/m2P,X ' (TPX)∗.

This can be viewed as an intrinsic definition of the cotangent space to X

at P, (TPX)∗, and consequently we have an intrinsic definition for the tangent

space to X at P :

(mP,X/m2P,X)∗ ' TPX.

In this way, we may define in general the so-called Zariski tangent space to

a scheme X at a point x, as the vector space

TxX = (mx,X/m2x,X)∗.


Remark 1.17. The tangent space TPX is an affine subspace of An passing

through the point P , but it has a natural structure of k-vector space by thinking

P as the zero vector and defining−→PQ = Q− P ∈ kn.

Since we may prove that always holds the inequality

dimkTPX ≥ dimPX

we say that P is nonsingular when we have the equality, otherwise we say that

P is singular.

In the case of affine subscheme of An, by elementary linear algebra, we have

that the dimension of TPX as k-vector space is exactly

dimkTPX = n− rank

[∂fi∂xj

]i=1,...,sj=1,...,n

.

Thus, a point P is nonsingular if and only if the rank of the Jacobian matrix

is

rank

[∂fi∂xj

]i=1,...,sj=1,...,n

= n− dimPX.

In the projective case, if X ⊂ Pn and P ∈ X, the first idea is to consider

the affine open set U ' An which contains the point P and take the closure in

the projective space of the well defined tangent space: we will continue to use

the notation TPX.

Assuming that the affine open is {x0 6= 0} and I(X) = (F1, . . . , Fs), we

may set

fi(x′1, . . . , x

′n) = Fi(1, x1/x0, . . . , xn/x0),

and hence X ∩Ui is the zero locus of (f1, . . . , fs). From the previous definition

we get that the affine tangent space at a point P = [1 : p1 : . . . : pn] ∈ X is the

locus

n∑j=1

∂f1

∂x′j(p1, . . . , pn)(x′j − pj) = . . . =

n∑j=1

∂fs∂x′j

(p1, . . . , pn)(x′j − pj) = 0.

1. Preliminaries 41

Thus, the projective tangent space, which is the closure in Pn of this locus, is

the zero locus of

n∑j=1

∂Fi∂xj

(1, p1, . . . , pn)(xj − pjx0) = 0, for all i = 1, . . . , s.

By the Euler relation for homogeneous polynomials of degree d

n∑i=0

∂F

∂xixi = dF,

so that, taking the polynomials F1, . . . , Fs, since Fi(1, p1, . . . , pn) = 0, we can

rewrite the equations of the tangent space which is the zero locus of

n∑i=0

∂fi∂xj

(P )xj = 0, for all i = 1, . . . , s.

After that, also in the projective case we may define singular and nonsin-

gular points in term of the Jacobian matrix exactly like the affine case.

In conclusion of this description of the tangent space, in the case of P a

nonsingular point of X ⊂ Pn, we may define the tangent space in terms of a

parametric representation of X in an analytic neighborhood of the point. If

we give a parametrization like

P (t1, . . . , ts) = [g0(t1, . . . , ts) : . . . : gn(t1, . . . , ts)] ∈ X

such that P (0) = P , the tangent space at P is the s-dimensional projective

space associated the the rows space of the matrixg0(0) . . . gn(0)

∂g0/∂t1(0) . . . ∂gn/∂t1(0)

. . . . . . . . .

∂g0/∂ts(0) . . . ∂gn/∂ts(0)

.

This definition will be very useful in the computations in the next chapters.

1.4 Secant Varieties 1. Preliminaries

1.4 Secant Varieties

In this section we are going to introduce the main topic of this thesis, the

secant varieties. From now, we assume k to be an algebraically closed field of

characteristic 0.

Definition 1.18. Let V a projective variety in Pn. We define the sth-higher

secant variety of V as the Zariski closure of the union of all (s−1)-dimensional

linear projective subspace spanned by s independent points of V , i.e.

σs(V ) :=⋃

P1,...,Ps∈Vindependents

〈P1, . . . , Ps〉.

Remark 1.18. The definitions can be viewed in the more general framework

of joins between projective varieties. Given two varieties X, Y in Pn, we define

the join between X and Y as the Zariski closure of the union of all lines spanned

by a point on X and a point on Y , i.e.

J(X, Y ) :=⋃

(P,Q)∈X×Y

〈P,Q〉.

Obviously, this definition can be extended to the join of several varieties

X1, . . . , Xs contained in the same Pn by considering the Ps−1’s spanned by s

points on each Xi, respectively.

From the definition of the join, the sth-secant variety of a variety V in Pn

can be defined as the join of s copies of V , i.e.

σs(V ) = J(V, . . . , V ).

Someone prefer to indicate this secant variety of V with the notation

Secs−1(V ), giving more importance to the dimension of the linear spaces used

for the definition rather than the number of points that span such spaces.

Directly from the definitions, we have a sequence of inclusions,

X = σ1(X) ⊂ σ2(X) . . . ⊂ σr(X) ⊂ . . . ⊂ Pn.

Without more information about the nature of our varieties, of course they

could be also equalities.

1. Preliminaries 43

Example 1.7. The first, and in some sense trivial, example is by taking X as

a linear subspace of a projective space. Obviously, we get that σi(X) = X for

all i ∈ N>0.

Remark 1.19. Let V be a variety in Pn and assume that σ2(V ) = V . Then,

V is a linear subspace in Pn.

Proof. Consider the span of V , denoted with 〈V 〉, i.e. the smaller linear sub-

space of Pn containing V . Let P ∈ 〈V 〉. There exist P1, . . . , Ps ∈ V such that

P is a linear combination of such points, i.e.

P = λ1P1 + . . .+ λsPs, where λi ∈ k.

Now, by definition, λ1P1 + λ2P2 ∈ σ2(V ) and then lies in V by hypothesis.

Consequently, λ1P1 + λ2P2 + λ3P3 ∈ V . In conclusion, we have proved that

P ∈ V , so that 〈V 〉 ⊂ V . Since always V ⊂ 〈V 〉, we are done.

Remark 1.20. More in general, let V be a variety in Pn and assume that

σi(V ) = σi+1(V ) 6= Pn for some i ≥ 2. In this case, we cannot say in general

that V is exactly a linear subspace of Pn. For example, if we consider a plane

curve, which is not a line, in P3, then it has clearly σ2(C) = σ3(C) equal to

the P2 in which it is contained, but it is not a linear subspace.

Claim : σi+h(V ) = σi(V ) for all h ≥ 0

and that V is contained in a proper linear subspace in Pn.

Proof. For the first fact, we just prove the first step for h = 2, but the same

argument works in general. Let P ∈ σi+2(V ) be a general point. By definition

P ∈ 〈P1, . . . , Pi+2〉, where Pi ∈ V , i.e. for some λi ∈ k

P = λ1P1 + . . .+ λi+1Pi+1 + λi+2Pi+2. (1.4)

Now, by hypothesis, there exist Q1, . . . , Qi ∈ V and µ1, . . . , µi ∈ k such that

λ1P1 + . . .+ λi+1Pi+1 = µ1Q1 + . . .+ µiQi. (1.5)


By replacing (1.5) in (1.4), we get that

P = µ1Q1 + . . .+ µiQi + λi+2Pi+2 ∈ σi+1(V )

and then we can say that σi+2(V ) ⊂ σi+1(V ). Since the converse inclusion is

always true, we can conclude the equality.

Now, we want to prove that if σi(V ) = σi+1(V ) for some i, then V is

contained in a projective linear space. Consider σ2(σi(V )) ⊂ Pn. The generic

point is of the type

P = λQ+ µR, where Q,R ∈ σi(V )

and then there exists Q1, . . . , Qi, R1, . . . , Ri ∈ V

P = λ1Q1 + . . .+ λiQi + µ1R1 + . . .+ µiRi, for some λj, µj ∈ k.

Hence P ∈ σ2i(V ) and then

σ2(σi(V )) ⊂ σ2i(V ).

Since the right hand side is equal to σi(V ) by the first part of this proof, we get

that the 2-secant variety of σi(V ) is equal to σi(V ). By Remark 1.19, σi(V ) is

a proper linear subspace in Pn. In conclusion, since σi(V ) 6= Pn, we can say

that V is contained in a proper linear subspace of Pn.

Definition 1.19. Let V be a variety in Pn. We say that V is degenerate if it

is contained in a proper linear subspace of Pn.

By Remark 1.20, if we assume that variety V is non-degenerate, we have

that the sequence of secant varieties of V has only strict inclusions, i.e.

V = σ1(V ) ( σ2(V ) ( . . . ( σr(V ) = Pn.

In summary, if V is non-degenerate

there exists a secant variety which fills the entire ambient space.

1. Preliminaries 45

Now, we can guess what is the first secant variety which should fill the

ambient space. Just counting parameters, we get that the expected dimension

of σs(V ) is

expdim(σs(V )) = min{s(dim(V )) + s− 1 , n}2

Hence, we expect that the first sth-secant variety which fills Pn is

s =

⌈n+ 1

dim(V ) + 1

⌉.

Unfortunately everything doesn’t work so easily in general. The actual

dimension of the sth-higher secant variety σs(V ) is always less or equal to the

expected one, but it is possible that it is strictly less. In such case, we say that

V is s-defective and the positive integer

δs(V ) := expdim(σs(V ))− dim(σs(V ))

is called the s-defect of V .

Remark 1.21. We can notice that, since always holds the inequality

dim σi+1V ≤ dim σiV + dim V + 1,

we get that if σiV 6= Pn and V is i-defective, then V is also j-defective for all

j ≥ i such that σjV doesn’t fill the ambient space.

Example 1.8. In order to see that the defective varieties actually exist also

among the classical families of projective varieties, we give an easy, and prob-

ably the most famous, example, the Veronese surface V 2,2 ⊂ P5. Just to recall

the definition, V 2,2 is the image of the Veronese embedding

ν2,2 : P2 −→ P5

[x0 : x1 : x2] 7−→ [x20 : x2

1 : x22 : x0x1 : x0x2 : x1x2] .

2The idea for this computation is that to determine a point on σs(V ) we have to find s

points on the variety V , i.e. we have s(dim(V )) degrees of freedom, and then a point on

their span, i.e. other s−1 degrees of freedom. Obviously, we have also to consider the upper

bound given by the dimension of the ambient space.


Now, consider the second secant variety of V 2,2. If we try to compute the

dimension just counting parameters, we need 4 degrees of freedom to determine

2 generic points on the surface and then one more degree of freedom to choose

a point on the line spanned by these two selected points. In summary, we have

expdim σ2(V 2,2) = 5.

In order to prove easily that V 2,2 is actually 2-defective, it is useful to recall

the determinantal representation of the Veronese surface in P5, i.e. as locus of

points [z0 : z1 : z2 : z3 : z4 : z5] ∈ P5 such that the matrix

M =

z0 z3 z4

z3 z1 z5

z4 z5 z2

has rank 1, or in other words, as the variety of matrices of rank 1 in the space

of symmetric matrices. Since a linear combination of two of such matrices has

rank at most 2, the first secant variety of the Veronese surface is contained in

the hypersurface of the equation det(M) = 0. Thus, it is defective. We will

see that the 2-defect of the Veronese surface really is 1, i.e. dim σ2(V 2,2) = 4.

The interest in secant varieties goes back to the beginning of XXth century

and especially to the Italian geometry school. For this reason we want to recall

at least the following classical theorem due to Palatini. Using such result we

will be able to say that curves are never defective.

Theorem 1.12 (Palatini, [Pal09]). Let V be a non-degenerate variety in Pn

and assume that dim(σs+1(V )) = dim(σs(V )) + 1. Then,

σs(V ) is an hypersurface in Pn.

Proof. Let x ∈ V and z ∈ σs+1V be general points, hence we have that

σs(V ) ( J(x, σs(V )) ⊂ σs+1(V ).

1. Preliminaries 47

By hypothesis, such inclusions force the equality J(x, σs(V )) ⊂ σs+1(V ) and

consequently we get that z ∈ 〈x, y〉 for some y ∈ σs(V ) and then x ∈ Tzσs+1(V ).

By generality of x ∈ V , this fact yields to

〈V 〉 ⊂ Tzσs+1(V ),

and then σs+1(V ) = Pn. Consequently, σs(V ) is an hypersurface in Pn.

In particular, we get that

dimσs+1(V ) ≥ dimσs(V ) + 2,

unless σs(V ) is an hypersurface in Pn.

Remark 1.22. Let C be a curve in Pn non-degenerate. If C is a plane curve

we know that σ2(C) fills the entire ambient space and consequently it is never

defective. In general, we have always the following upper bound

dim σs+1(C) ≤ dim σs(C) + 2.

Since C is not an hypersurface, by Palatini’s Theorem, we get that, unless σs(C)

is an hypersurface,

dim σs+1(C) ≥ dim σs(C) + 2.

Consequently, we get the equality and we can say that non-degenerate curves

are never defective.

Even though the problem of defectiveness of projective varieties has a quite

long story, it is largely unknown. Up to now, the problem has been attacked

case by case, firstly considering the more classical varieties like Veronese, Segre,

Segre-Veronese and Grassmannians.

Unfortunately, in this moment, the unique case completely solved is the

one of Veronese varieties. In the following chapter we will discuss about this

case, also trying to give an idea about the methods used by Alexander and

Hirschowitz in 1995 to complete the classification of defectiveness of Veronese

varieties.

Chapter 2

Veronese varieties

In this chapter we want to talk about the unique family of projective vari-

eties for which we have a complete classification in term of defectiveness: the

Veronese varieties.

The history of this problem is long more than a century and involved a lot

of mathematicians. The first approach was by considering some specific case

and, since the Veronese varieties parametrize the powers of linear forms, it was

very algebraic. For example, we can find some results in several papers written

by Clebsch, Cambpell and Richmond in the last years of XIX century.

Meanwhile, also the Italian school of geometry of Corrado Segre began to

study such kind of issues. Palatini attacked the problem in a more geometrical

way, but the turning point of the story is due to Terracini.

In his famous paper of 1915 [Ter15], he introduced new techniques which

simplify the approach to the problem, even if he was far away from the final

solution.

The last key point in the story of defectiveness of Veronese varieties is due to

Hirschowitz in 1985 [Hir85]. He gave an easy proof of the specific cases already

known by using the powerful language of 0-dimensional schemes. After that,

in a long joint work with Alexander, they improved such method until the

celebrated paper of 1995 [AH95] where they gave the complete classification of

defective Veronese varieties: the so-called Alexander-Hirschowitz Theorem.

49

2. Veronese varieties

We will try to explain the methods used in the proof of the Alexander-

Hirschowitz Theorem. The first step will be to explain the relation between

secant varieties of Veronese varieties and 0-dimensional schemes of Pn. Firstly,

we will explain it by using the language of inverse system of fat points which is

not the classical approach to the problem, but it can be useful the study more in

general problems dealing with the secant varieties of Veronese varieties. Later

we will describe also the classical geometrical approach by using the Terracini’s

Lemma.

2.0.1. Preliminaries. Let us recall the definition of Veronese varieties.

Definition 2.1. Let n and d be two positive integers. The Veronese variety

V n,d is defined as the image of the Veronese embedding

νn,d : Pn −→ PN

[x0 : . . . : xn] 7−→ [xd0 : xd−10 x1 : . . . : xdn]

(2.1)

where N =(n+dn

)− 1.

This map can be viewed in terms of the symmetric algebra SV associated

to a k-vector space V of dimension n + 1. The Veronese variety V n,d is the

image of

νn,d : P(V ) = P(S1V ) −→ P(SdV )

[L] 7−→ [Ld]. (2.2)

This point of view is useful to relate the problem of defectiveness of Veronese

varieties also to the Waring problem for polynomials.

Let S = k[x0, . . . , xn] ' SV .

“Little” Waring problem: let d be a positive integer, what is the minimal

integer g(d) such that all the homogeneous polynomials of degree d can

by written as sum of ≤ g(d) linear forms?

2. Veronese varieties 51

“Big” Waring problem: let d be a positive integer, what is the minimal

integer G(d) such that the generic homogeneous polynomials of degree d

can by written as sum of ≤ G(d) linear forms?

From the description (2.2) of the Veronese embedding, it is clear that the

Veronese variety V n,d parametrized the set of dth-powers of linear forms, thus

the generic point on the secant variety σs(Vn,d) corresponds to a linear combi-

nation of s powers of linear forms and, since k is algebraically closed, to a sum

of powers of linear forms. This relation was one of the main reasons for what

a lot of mathematicians began to study this kind of problems.

By Alexander-Hirschowitz Theorem, we can solve the “Big” Waring’s prob-

lem. The knowledge of defectiveness of Veronese varieties let us able to say

when σs(Vn,d) fills the space in which it is embedded and hence, when the

generic form of degree d can be written as sum of s powers of linear forms. To

solve the “Little” Waring’s problem is quite harder because we should have a

greater knowledge of the secant varieties, point by point, and it is, in general,

a still unsolved problem.

In the next section, we are going to connect the problem dealing with

the defectiveness of Veronese varieties to the Hilbert function of 2-fat points

which actually is the setting where Alexander and Hirschowitz have stated

their theorem. To do this, we can use algebraic tools called inverse system and

apolarity, but a more classical and geometrical approach would be by using the

“Terracini’s Lemma”. We will see both of them.

2.1 Hilbert function of fat points via apolarity

In order to understand the spirit of the problem, which can be viewed as an

interpolation problem, we begin by considering the case of s simple points,

X = {P1, . . . , Ps}, i.e. a reduced 0-dimensional subscheme in Pn.

Let ℘i ∈ Proj k[x0, . . . , xn] = Proj S be the primes ideals of height n which

2.1 Hilbert function of fat points via apolarity 2. Veronese varieties

correspond, respectively, to the point Pi, i.e.

℘i = (Li,1, . . . , Li,n) where Li,j ∈ S1;

hence the ideal corresponding to X is the intersection

I = ℘1 ∩ . . . ∩ ℘s.

Recalling the definition of Hilbert function of a graded module given in the

Section 1.1, the Hilbert function of X is defined as

HFX(t) := dim (St/It) .

Let M1, . . . ,M(d+nn ) be the standard monomial basis of Sd as k-vector space,

then we can write an element of Sd as

F = c1M1 + . . .+ c(d+nn )M(d+nn ),

where ci ∈ k.

So, the forms of degree d which vanish at the points P1, . . . , Ps, are precisely

the solutions of the linear systemM1(P1)c1 + . . . + M(d+nn )(P1)c(d+nn ) = 0

......

......

M1(Ps)c1 + . . . + M(d+nn )(Ps)c(d+nn ) = 0

(2.3)

which we can write as

Md ·

c1

...

c(d+nn )

= 0.

Since the k-vector space Id is the set of the solutions of the system (2.3),

its dimension can be computed as

dim (Id) =

(d+ n

n

)− rank(Md),

thus we get

HFS/I(d) =

(d+ n

n

)− dimk(Id) = rank(Md).


Since we can pick points P1, . . . , Ps such that the matrices Md all have

maximal rank, we get that

a general set X of s points in Pn has Hilbert function

HFX(t) = min

{s,

(t+ n

n

)}.

Now, the idea is to extend this kind of interpolation problem to the case of

non-reduced 0-dimensional schemes, in particular we are going to investigate

about the Hilbert function of families of 2-fat points.

Following the common spirit which says that “firstly you try with the sim-

plest case”, we begin by assuming that our subscheme of Pn is supported at a

single point P . In order to set ideas, let P = [1 : 0 : . . . : 0], i.e. we are looking

at a subscheme associated to an ideal with radical ℘ = (x1, . . . , xn).

Firstly, let us explain the reason of the classical interest in these ideals. Let

F ∈ ℘ be an homogeneous polynomial of degree d.

If we dehomogenize F with respect the variable x0, we obtain f ∈ k[x1, . . . , xn],

by setting f(x1, . . . , xn) = F (1, x1/x0, . . . , xn/x0), and the point P becomes the

origin 0 = (0, . . . , 0) ∈ An, we continue to denote it as P . So we can write

f = f0 + . . .+ fd where deg(fi) = i

and moreover, since F ∈ ℘, we have f(P ) = 0, i.e. f0 = 0.

After that, since

f1 = ((∂f/∂x1)|0)x1 + . . .+ ((∂f/∂xn)|0)xn,,

if all the first partial derivatives of f do not vanish at P = 0, we say that P

is a smooth point of V (f) and f1 = 0 is the equation of the tangent hyperplane

at P . Hence, we can say that:

F ∈ ℘r ℘2 ⇔ at least one of the first partial derivatives of f

does not vanish at P

or putting in another way:

F ∈ ℘2 ⇔ (∂F/∂xi)|0 = 0 for all i = 1, . . . , n,

⇔ P is a singular point of V (F ).


So, if I = ℘2 then Id consists of all the forms of degree d which have a

singularity at P . Proceeding in this way, by taking the entire Taylor expansion

of f around P = 0, we can notice that

F ∈ ℘t ⇔ all the partial derivatives of f of order ≤ t− 1 vanish at 0

⇔ P is a singular point of V (F ) with multiplicity ≥ t.

Therefore, we can say that

F ∈ ℘t and deg(F ) = d if and only if f = ft + . . .+ fd.

As a consequence of these facts, we get

HF(S/℘t, s) =

(s+nn

)if s < t(

t−1+nn

)if s ≥ t

Remark 2.1. From the previous observations, a single t-fat point tP in Pn,

from the point of view of the Hilbert functions, “behaves like”(t−1+nn

)distinct

points of Pn in the sense that for s� 0 they have the same Hilbert function.

Now, we would like to extend these kind of arguments to the case of a

generic subscheme of fat points in Pn. Let P1, . . . , Ps be distinct points in

Pn with ideals ℘1, . . . , ℘s, respectively. If m1, . . . ,ms are positive integers, we

consider the subscheme m1P1 + . . .+msPs of Pn defined by the ideal

I = ℘m11 ∩ . . . ∩ ℘mss .

Our target is to compute the Hilbert function of S/I which, as we have

already said, can be viewed like a differential interpolation problem. We are

asking which is the dimension of the space of hypersurfaces of a given degree

which pass through a given set of points and have, at those points, a singularity

of multiplicity at least mi.

Consequently to the previous argument made in the case of a single fat

point, one could expect that, for sufficiently general set of points, I has the

same Hilbert function of∑s

i=1

(s−1+nn

)distinct points in Pn.

Unfortunately it is easy to find some counterexample to this conjecture:


Example 2.1. Let P1 and P2 any two points in P2 and consider I = ℘21 ∩ ℘2

2.

Then if the above assumption is true, I should have the Hilbert function of

6 general points in P2 which is

1 3 6 6 . . .

so there should be no conic in I. But, if L is the line through P1 and P2, then

L2 ∈ I and so HF{P1,P2}(2) < 6.

Similarly, let P1, . . . , P5 be 5 general points in P2 and consider I = ℘21 ∩

. . . ∩ ℘25. It should have the same Hilbert function of 15 general points in P2,

which is

1 3 6 10 15 15 . . .

then there should be no quartic in I. But, if C is the conic through such 5

points, then C2 ∈ I and so HF{P1,...,P5}(4) < 15.

2.1.1. Inverse System. Now, we are going to introduce an algebraic tool

which can be very useful to study the 0-dimensional subschemes:

the Macaulay’s Inverse System.

In our case, it will allow us to relate the problem concerning secant varieties

of Veronese varieties to the Hilbert function of 2-fat points.

Let’s to consider two polynomial rings at the same time:

S = k[x1, . . . , xn] and R = k[y1, . . . , yn];

we think of S as representing the partial differential operators on the poly-

nomials in R. The action of S on R is called apolarity and it is defined by

setting

xi ◦ yj =∂

∂yi(yj) =

0 if i 6= j

1 if i = j.

,


hence, S1 can be thought like the dual space of R1. Using the standard

properties of differentiation, we can extend this action to

Si ×Rj −→ Rj−i

(si, rj) 7−→ s ◦ r(2.4)

Moreover, we can notice that this action makes R into an S−module, i.e.

(i) s ◦ (r1 + r2) = s ◦ r1 + s ◦ r2;

(ii) (s1s2) ◦ r = s1 ◦ (s2 ◦ r);

(iii) (s1 + s2) ◦ r = s1 ◦ r + s2 ◦ r;

(iv) 1 ◦ r = r;

(v) s ◦ (cr) = (cs) ◦ r = c(s ◦ r), with c ∈ k.

Example 2.2. Let F2 = x21 + x1x2 and G4 = y4

1 + y42. Then,

F2 ◦G4 = 12y21 ∈ R2.

Now, if we write a monomial in the short form

xα := xα11 · . . . · xαnn , where α = (α1, . . . , αn) ∈ Nn,

by a simple computation, we get that 1

xα ◦ yβ =

0 if α 6≤ β∏ni=1((βi)!/(βi − αi)!)yβ−α if α ≤ β

(2.5)

From the properties above, the apolarity action induces a k−bilinear pairing

Si ×Ri → k,

and, thanks to the description (2.5) of the apolarity action, we get that if

we order the standard monomial basis, the matrix associated to such bilinear

1For each α, β ∈ Nn, we say α ≤ β ⇔ αi ≤ βi for all i = 1, . . . , n.


pairing is a diagonal matrix with nonzero entries on the diagonal; thus it is a

nonsingular bilinear pairing2.

So, if V ⊆ Si and W ⊆ Ri are two k−vector subspaces, we can easily define

their “perp”:

V ⊥ = {f ∈ Ri | v ◦ f = 0 for all v ∈ V } ,

W⊥ = {g ∈ Si | g ◦ w = 0 for all w ∈ W} .

Therefore, since it is a nonsingular pairing, we get that

dimk(V⊥) = dimk(Si)− dimk(V ).

Definition 2.2. Let I be an homogeneous ideal of the ring S. The inverse

system of I, denoted I−1, is the S−submodule of R consisting of all elements

of R annihilated by I with the apolarity action.

Example 2.3. Let I = (x1) ⊆ S = k[x1, x2]. By definition,

I−1 = {G ∈ R | (∂/∂y1)(G) = 0} .

Now, since also I−1 is graded, to compute I−1 we find what it looks like in

every degree.

Let ay1 + by2 ∈ R1, then

x1 ◦ (ay1 + by2) = a. Thus, (I−1)1 = 〈y2〉.

Let ay21 + by1y2 + cy2

2 ∈ R2, then

x1 ◦ (ay21 + by1y2 + cy2

2) = 2ay1 + by2. Thus, (I−1)2 = 〈y22〉.

Proceeding in this way, it easily follows that

I−1 = k ⊕ 〈y2〉 ⊕ 〈y22〉 ⊕ 〈y3

2〉 ⊕ . . .

The following proposition will help us to compute the inverse system of an

ideal I more in general:

Proposition 2.1.

(I−1)j = I⊥j , for all j ∈ N.

2We say that a pairing is nonsingular if the associated matrix is nonsingular.


Proof. By definition

Ij × I⊥j → 0,

i.e.

(I−1)j ⊆ I⊥j .

Conversely, suppose G ∈ I⊥j : we have that h ◦G = 0 for all h ∈ Ij and we

need to prove that F ◦G = 0 for all F ∈ I.

In the case deg(F ) > j, F ◦G = 0 simply because the degree of F is greater

than the degree of G.

In the case deg(F ) < j, let α = (α1, . . . , αn) such that

n∑i=0

αi = j − deg(F ).

Then deg(xαF ) = j and xαF ∈ Ij. Thus (xαF ) ◦ G = xα ◦ (F ◦ G) = 0, for

all monomial xα. Since deg(xα) = j − deg(F ) and F ◦G ∈ Rj−deg(F ), and the

apolarity pairing is nonsingular we get F ◦G = 0.

This proposition has a direct, but very interesting, consequence:

Corollary 2.2.

dimk(I−1)j = dimk(Sj/Ij) = HFS/I(j), for all j ∈ N.

Proof. Since I−1j = I⊥j and the apolarity pairing is nonsingular,

dimk(I⊥j ) = dimk(Rj)− dimk(Ij) = dimk(Sj)− dimk(Ij).

This is a useful connection because we can convert the computation of

the Hilbert function of an ideal to a discussion about the dimension of the

homogeneous parts of the inverse system of an ideal.


Remark 2.2. Proposition 2.1 gives us a simple way to describe the inverse

system in the monomial case. In fact, if I is a monomial ideal, we have I−1j = I⊥j

and we know exactly how I⊥j looks like:

I⊥j = 〈the monomials which are not in Ij〉.

There is also an easy way to compute the inverse system of an intersection

of ideals:

Proposition 2.3. Let I and J two ideals:

(I ∩ J)−1 = I−1 + J−1.

Proof. The proof is a direct consequence of the following easy linear algebra

fact. Let V ×W → k be a nonsingular pairing with dimk(V ) = dimk(W ), if

U1 and U2 are two subspaces of V , then

(U1 ∩ U2)⊥ = U⊥1 + U⊥2 .

2.1.2. Inverse System of Fat Points. After these general definitions, we

go back to our previous assumptions and we restart to talk about the Hilbert

function of fat points in Pn.

Firstly we focus our attention to the case of a singular fat point supported

on P ∈ Pn and associated to the ideal ℘ = (L1, . . . , Ln). We can make a linear

change of variables in Pn so that

P = [1 : 0 : . . . : 0] and ℘ = (x1, . . . , xn).

Let I = ℘t+1. From the remark above on the inverse system in the monomial

cases, we get that

I−1 = 〈yβ | xβ /∈ I〉.

Since Ij = (0) for j ≤ t, then

(I−1)j = Rj for j ≤ t.


Now, we may write 3

[(℘t+1)−1

]i

= 〈yi0〉 ⊕ 〈yi−10 T1〉 ⊕ . . .⊕ 〈yi−t0 Tt〉 =

= yi−t0 Rt.

Formally we get

(℘t+1)−1 = R0 ⊕R1 ⊕ . . .⊕Rt ⊕ y0Rt ⊕ y20Rt ⊕ . . .

Now, suppose that P is a general point in Pn, say P = [p0 : . . . : pn] where,

without loss generality, we may assume p0 = 1. Thus,

℘ = (x1 − p1x0, . . . , xn − pnx0).

By using the following change of variables,

x′0 = x0

x′1 = x1 − p1x0

...

x′n = xn − pnx0

we have, like above,

[(℘t+1)−1

]i

=

(y′0)i−tRt if i ≥ t

Ri if i < t

If we consider the inverse of such change of variables, we get

x′0...

x′n

= A ·

x0

...

xn

where A =

1 0 0 . . . 0

−p1 1 0 . . . 0

−p2 0 1 . . . 0...

.... . .

......

−pn 0 0 . . . 1

,

and hence, for the dual basis,y′0...

y′n

=(A−1

)t ·y0

...

yn

.3For convenience we set T = k[y1, . . . , yn].


So, we obtain

y′0 = y0 + p1y1 + . . .+ pnyn

y′1 = y1

...

y′n = yn

In summary, we get the following results

Lemma 2.4. If P = [p0 : . . . : pn] ∈ Pn and ℘ the associated ideal, then

(℘t+1)−1 = R0 ⊕R1 ⊕ . . .⊕Rt ⊕ LRt ⊕ L2Rt ⊕ . . .

where L = p0y0 + . . .+ pnyn.

By Lemma 2.4 and Proposition 2.3, we get the following important theorem:

Theorem 2.5 (Apolarity Lemma). Let P1, . . . , Ps be distinct points of Pn

and suppose Pi = [pi0 : . . . : pin]. Let

Li = pi0y0 + pi1y1 + . . .+ pinyn ∈ R = k[y0, . . . , yn].

Then, if I = ℘n1+11 ∩ . . . ∩ ℘ns+1

s ⊆ S = k[x0, . . . , xn] we have

(I−1)j =

Rj for j ≤ max{ni}

Lj−n1

1 Rn1 + . . .+ Lj−nss Rns for j ≥ max{ni + 1}

Moreover, by Corollary 2.2 we also have the following result about Hilbert

function of set of fat points.

Corollary 2.6. Let I = ℘m1+11 ∩ . . . ∩ ℘ms+1

s ⊆ S = k[x0, . . . , xn] an ideal as

above. Then

HFS/I(j) = dimk(I−1)j =

=

dimk(Sj) for j ≤ max{mi}

dimk〈Lj−m1

1 Rm1 , . . . , Lj−mss Rms〉 for j ≥ max{mi + 1}


We can notice that, with these last results, there is a strong relationship

between the Hilbert function of fat points and the ideals generated by powers

of linear forms. We are going in the right direction to find a connection to the

Waring’s problem for polynomials and then to the defectiveness of Veronese

varieties. More precisely, Theorem 2.5 says that

(I−1)j is the jth graded piece of the ideal (Lj−m1

1 , . . . , Lj−mss )

for j ≥ max{mi + 1}.

2.1.3. Secant Varieties, Fat Points and Big Waring’s problems.

Finally, we are going to explain which is the connection between the top-

ics that we have introduced in the previous paragraphs: secant varieties of

Veronese varieties, Hilbert function of fat points and Waring problem for poly-

nomials.

Recall the “Big” Waring problem for polynomials:

which is the minimal integer G(d) such that the generic form

F ∈ Rd = k[y0, . . . , yn]d can be written

as sum of at most G(d) dth powers of linear forms?

Geometrically speaking, we have seen that it can be translated to the problem:

which is the minimal integer G(d) such that the G(d)th-secant varieties of the

Veronese variety V n,d fills all space in which it is embedded?

In other words, we need to understand what is the dimension of the image

of the map

φ : R1 × . . .×R1 −→ Rd

(L1, . . . , Ls) 7−→ Ld1 + . . .+ Lds


Since we can see R1 like An+1 and Rd as AN(n,d)+1, where N = N(n, d) =(n+dn

)− 1, we can see that map as

φ : As(n+1) −→ AN ,

and since we are interested to the image’s dimension, we take the differential

dφ which is a function such that, for each point P ∈ As(n+1), gives a linear

transformation between the tangent spaces

dφ(P ) := dφ|P : TP(As(n+1)

)−→ Tφ(P )

(AN).

Since the tangent space to At at any points is again At, we have that

dφ(P ) := dφ|P : As(n+1) → AN .

Thus, if we know the generic rank of these linear transformations, we will

know the dimension of the image, so that our problem becomes a differential

geometry problem:

given a vector v ∈ TP(As(n+1)

), how do we find dφ|P (v)?

The usual way is to consider a curve C through the point P , whose tangent

vector is v, and then take the image φ (C) and compute its tangent vector at

φ(P ).

Let P = (L1, . . . , Ls) ∈ As(n+1) and v ∈ TP(As(n+1)

)= As(n+1), so we may

assume v = (M1, . . . ,Ms), with Mi are elements in R1. The parametrized

curve in As(n+1) through P and with tangent vector v at P could simply be

C : t→ (L1 + tM1, . . . , Ls + tMs) .

The image of that curve C via the map φ is

φ (C) =s∑i=1

(Li + tMi)d.

Thus, we can compute the tangent vector in AN at the point φ(P ):

d

dt

∣∣∣∣t=0

(s∑i=1

(Li + tMi)d

)=

(s∑i=1

d (Li + tMi)d−1Mi

∣∣∣∣∣t=0

=s∑i=1

dLd−1i Mi.


Since we let v = (M1, . . . ,Ms) vary over the whole space As(n+1), the im-

age varies over the vector space⟨Ld−1

1 R1, . . . , Ld−1s R1

⟩, then the rank of the

differential at P = (L1, . . . , Ls) is equal to dimk

⟨Ld−1

1 R1, . . . , Ld−1s R1

⟩.

Considering this last observation and Corollary 2.6 proved in the previous

paragraph, we obtain:

Theorem 2.7. Using notations as above:

let L1, . . . , Ls be generic linear forms in R = k[y0, . . . , yn] where

Li = ai0y0 + . . .+ ainyn;

let P1, . . . , Ps be the corresponding generic points in Pn, namely

Pi = [ai0 : . . . : ain];

let ℘1, . . . , ℘s be the corresponding prime ideals in R, namely

℘i = (x0 − ai0, . . . , xn − ain).

If we consider the function φ(L1, . . . , Ls) = Ld1 + . . .+ Lds :

rank(dφ)|(L1,...,Ls)= dimk

⟨Ld−1

1 R1, . . . , Ld−1s R1

⟩= HFR/℘2

1∩...∩℘2s(d).

Now, recalling that the variety in PN which is the closure of the set of

forms in Rd which are sum of at most s dth-powers of linear forms, is the sth-

secant varieties of the Veronese variety V n,d in PN , the last theorem explains

the connection between such secant varieties and the Hilbert function of 2-fat

points which can be rephrased as follows.

Theorem 2.8. Let P1, . . . , Ps ∈ Pn be a set of generic points and let ℘1, . . . , ℘s

be the corresponding prime ideals in R. Then

dim(σs(V

n,d))

= HFR/℘21∩...∩℘2

s(d)− 1 = dimk

(Rd

[℘21 ∩ . . . ∩ ℘2

s]d

)− 1.

Now, we can enounce the theorem of J.Alexander and A.Hirchowitz in the

original framework.

Theorem 2.9 (J.Alexander, A.Hirschowits). Let X = {P1, . . . , Ps} be

a set of general s distinct points in Pn, let ℘i ⊆ R = k[y0, . . . , yn] be the


associated ideals. Then

HFR/℘21∩...∩℘2

s(d) = min {s(n+ 1), dimk Rd} ,

except for

(i) d = 2, 2 ≤ s ≤ n;

(ii) n = 2, d = 4, s = 5;

(iii) n = 3, d = 4, s = 9;

(iv) n = 4, d = 4, s = 14;

(v) n = 4, d = 3, s = 7.

In term of defectiveness of Veronese varieties, by Theorem 2.8, the Alexander-

Hirschowitz Theorem may be rephrased as follows.

Corollary 2.10. The dimension of the sth-secant variety of the Veronese va-

riety V n,d ⊂ PN is

dim(σs(V

n,d))

= min

{s · n+ s− 1,

(n+ d

d

)− 1

},

with the numerical exception of the Theorem 2.9.

2.2 Terracini’s Lemma

The classical approach to the problem dealing with the secants varieties is based

on the famous work of A. Terracini of 1915. Using the so-called Terracini’s

Lemma in the particular case of the Veronese varieties we may describe in

a more geometrical way the connection between the Hilbert function of 2-fat

points and the dimension of the secants of Veronese varieties.

The reason of this section is that the Terracini’s Lemma is completely gen-

eral so that it is a fundamental tool for studying the secant varieties of projec-

tive varieties in general.

2.2 Terracini’s Lemma 2. Veronese varieties

Lemma 2.11 (Terracini’s Lemma). LetX be an irreducible non-degenerate

projective variety and let P1, . . . , Ps ∈ X be a family of general points. If

z ∈ 〈P1, . . . , Ps〉 is general, then

Tz σs(X) = 〈TP1X, . . . , TPsX〉.

Proof. The thesis follows with a direct computation of the two spaces.

Let X(τ) = X(τ1, . . . , τn) be a parametrization of X. So, we get Pi =

X(τ(i)1 , . . . , τ

(i)n ) = X(τ (i)), for all i = 1, . . . , s, and then the tangent space

TPiX is generated by the rows of the matrix

Ti =

X(τ

(i)1 , . . . , τ

(i)n )

∂∂τ1X(τ

(i)1 , . . . , τ

(i)n )

...

∂∂τnX(τ

(i)1 , . . . , τ

(i)n )

.Hence, the right-hand side of the thesis is generated by the s(n + 1) rows

of the matrix construct by taking all such matrix Ti in column.

On the other hand, we can express a parametrization for σs(X):

σs(X)(τ (1), . . . , τ (s);λ1, . . . , λs−1) =s−1∑i=1

λiX(τ (i)) +X(τ (s)).

Hence, the tangent space of σs(X) at z is the rows space of the following

matrix

∑s−1i=1 λiX(τ (i)) +X(τ (s))

λ1∂

∂τ(1)1

X(τ (1))

...

λi∂

∂τ(i)j

X(τ (i))

...

X(τ 1)...

X(τ s−1)

.

This is also a matrix of s(n + 1) rows and it is easy to check that one can be

obtained from the other by elementary row operations and then, they span the

same row space.


Remark 2.3. Using the Terracini’s Lemma, we can give a geometrical justi-

fication of defectiveness of the second secant variety of the Veronese surface

V 2,2 in P5. We have already talked about it, using a completely algebraic ar-

gument, in Example 1.8, but this approach allows us to better understand the

“geometrical reason” of this defectiveness and also which is the defect.

As we have already computed in Section 2.1.3, the tangent space to a point

P = [L2] ∈ V 2,2 is

TPV2,2 = {[LM ] | M ∈ S1}.

Taking a generic point A on the secant variety σ2(V 2,2) which lies on the span

of two generic points on the Veronese surface, P = [L2] and Q = [N2], by

Terracini’s Lemma, we get

TAσ2(V 2,2) = 〈TPV 2,2, TQV2,2〉.

As we have largely understood, we need to compute the dimension of such

tangent space, so we will compute the dimension of intersection of the two tan-

gent space to the Veronese surface at P and Q in order to use the Grassmann’s

formula. Since P and Q are generic points, the associated linear forms L and

N are independents, thus it immediately follows that

TPV2,2 ∩ TQV 2,2 = [LN ] ∈ P5.

By the Grassmann’s formula

dim TAσ(V 2,2) = 3 + 3− 1− 1 = 4,

and hence σ2(V 2,2) is a hypersurface and does not fill the entire space P5 as we

expected.

As we have already said, we want to find again the connection between

Hilbert function of fat points and secant varieties of Veronese varieties, but

using a geometrical approach which is based on the Terracini’s Lemma and the

geometrical nature of the Veronese embedding νn,d : Pn → PN .

2.2 Terracini’s Lemma 2. Veronese varieties

Remark 2.4. If H ⊂ PN is a hyperplane, then ν−1n,d(H) ⊂ Pn is an hypersurface

of degree d. The key point is that if x0, . . . , xn are the coordinates of Pn, then

the coordinates z0, . . . , zN in PN are the monomial of degree d in the xi. In

this way, if H has equation a0z0 + . . . + aNzN = 0, the equation of ν−1n,d(H) is

a0xd0 + a1x

d−10 x1 + . . .+ aNx

dn = 0.

Remark 2.5. Let H ⊂ PN be an hyperplane and [Ld] ∈ H, since ν−1n,d([L

d]) =

[L], then we have that ν−1n,d is an hypersurface of degree d in Pn passing through

[L]. More precisely, we can observe that if we get H ⊃ T[Ld]Vn,d, then ν−1

n,d(H)

is an hypersurface singular at [L].

Proof. Without loss of generality, we may assume Ld = xd0, or equivalently

P = [Ld] = [1 : 0 : . . . : 0] ∈ PN . The generic hyperplane H passing through P

has equation

a1z1 + . . .+ aNzN = 0,

so that the hypersurface ν−1n,d(H) has equation

a1xd−10 x1 + . . .+ aNx

dn = 0,

which passes through the point ν−1n,d(P ) = [1 : 0 : . . . : 0] ∈ Pn.

As we have seen previously, the tangent space TPVn,d is the linear span of

the forms

xd0, xd−10 x1, . . . , x

d−10 xn,

and hence of the first n+ 1 basic point of PN

[1 : 0 : . . . : 0], . . . , [0 : . . . : 1 : . . . : 0].

Thus, if H ⊃ TPVn,d, the corresponding hypersurface in Pn has the equation

an+1xd−20 x2

1 + . . .+ aNxdn = 0,

which is singular at the point [1 : 0 : . . . : 0] ∈ Pn.

Using this fact, we get that there exist a bijection between

{hyperplanes H ⊂ PN such that H ⊃ TP1Vn,d, . . . , TPsV

n,d, for Pi ∈ V n,d}

and

{hypersurfaces of degree d in Pn singular at ν−1n,d(P1), . . . , ν−1

n,d(Ps)} =

= [℘21 ∩ . . . ∩ ℘2

s]d,


where ℘i is the ideal defining ν−1n,d(Pi).

Finally, from the Terracini’s Lemma and this last bijection, since the span

of some linear spaces is the smallest linear space which contains all of them,

or equivalently the intersection of all hyperplane which contain all of them, we

get the following result:

Lemma 2.12. Let V n,d ⊂ PN and choose generic points P1, . . . , Ps ∈ Pn with

defining ideals ℘1, . . . , ℘s respectively. Then

dim σs(Vn,d) = N − dim[℘2

1 ∩ . . . ∩ ℘2s]d.

Since dim Sd = N + 1, we can rephrase such result in term of Hilbert

function and then we get again the Theorem 2.8:

dim σs(Vn,d) = HFS/℘2

1∩...∩℘2s(d)− 1.

2.3 Alexander-Hirschowitz Theorem

Finally, in this section we are going to explain in more details the Alexander-

Hirschowitz Theorem. The first step will be to prove the exceptional cases,

after that we will try to give an idea about the method that allowed them to

demonstrate the theorem in the general case.

Definition 2.3. Let X ⊂ Pn be a generic set of s 2-fat points, we say that X

imposes independent conditions on the linear system of forms of degree d in

n+ 1 variables OPn(d) if the following equality holds

codim IX(d) = HFX(d) = min{s(n+ 1), dimk Sd}.

From Theorem 2.8, we can immediately see that the problem dealing with the

conditions imposed by a set of double points on the hypersurfaces of certain

degree in Pn is just a different way to see the problem of defectiveness of secant

varieties of Veronese varieties.

2.3 Alexander-Hirschowitz Theorem 2. Veronese varieties

Remark 2.6. Let X be a set of s double points in Pn in general position.

Assume that there are no hypersurfaces of degree d with such singularities,

i.e. the dimension of IX(d) is zero. Clearly, for any set of double points X′

which contains X the dimension of IX′(d) is again zero. In therms of secant

varieties, this is equivalent to say that if σs(Vn,d) fills the entire ambient space,

the following secant varieties σs+h(Vn,d) again fill the ambient space for any

h ∈ N.

Instead, assume that the codimension of X is equal to s(n + 1), i.e. such

double points impose as much conditions on OPn(d) as the degree of X. For

any set of s′′ double points X′′ contained in X, the codimension of IX′′(d) is

equal to the degree s′′(n + 1). In terms of secant varieties, it is the same to

say that if σs(Vn,d) does not fill the ambient space and it is non defective, then

the secant varieties σs−h(Vn,d) for any h = 0, . . . , s − 1 are not defective. See

Remark 1.21.

Remark 2.7. Let X = {2P1. . . . , 2Ps} be a set of s general double points

which impose independent conditions on the linear system OPn(d) and moreover

assume that (s + 1)(n + 1) <(n+dn

). In other words we have a set of double

points X which impose as much conditions as the degree and moreover “there

is enough space” for adding one more point Q and asking if Q imposes other

n+ 1 independent conditions on OPn(d).

In order to prove it, we just have to find n+ 1 independent forms of degree

d such that

· one of them is singular at P1, . . . , Ps, but doesn’t pass through Q;

· for any independent direction in Pn we can find a form of degree d which is

singular at P1, . . . , Ps and in Q has the tangent with such direction.

This is a very useful fact to understand when adding a double point we are

actually imposing n + 1 more conditions on our linear system. Such result is

been generalized by Chandler with the Curvilinear Lemma 2.19 that we will

see without proof in the next chapter.


2.3.1 Terracini’s Second Lemma

In order to understand some classic methods to approach this kind of problem,

we are going to prove another result due to Terracini which will allow us to

prove the Alexander-Hirschowitz Theorem in the special case of set of double

points in the projective plane P2.

Lemma 2.13 (Second Terracini’s Lemma). Let V be a variety in PN .

Assume that V is s-defective. Then, if {P1, . . . , Ps} is a set of general points

on V we get that

there exists a positive dimensional variety C ⊂ V passing through all Pi

and such that if P ∈ C then TPV ⊂ 〈TP1V, . . . , TPsV 〉.

Proof. Let A be a general point in 〈P1, . . . , Ps〉. By Terracini’s Lemma 2.11,

we have

TA σsV = 〈TP1V, . . . , TPsV 〉 .

Now we may observe that the secant variety σsV can be obtained by projecting

on the last factor the abstract secant variety

σs V := {(Q1, . . . , Qs, A) | A ∈ 〈Q1, . . . , Qs〉, dim〈Q1, . . . , Qs〉 = s− 1 }

⊂ V × . . .× V × PN .

which has dimension equal to ns+ s− 1.

By assumption, the s-secant variety has not the expected dimension and

then the fibers QA of the above projection have positive dimension, moreover

they are invariant under permutations. Note that (P1, . . . , Ps) ∈ QA and also

z ∈ 〈Q1, . . . , Qs〉 for all (Q1, . . . , Qs) ∈ QA such that dim〈Q1, . . . , Qs〉 = s− 1.

In particular, for any such Q1, we have

TQ1σsV ⊂ 〈TP1V, . . . , TPsV 〉 .

Hence, the image of QA on the first component is the variety C that we

looked for.


In the previous section, we have seen that if an hyperplane H in PN con-

tains the tangent space at a point P of the Veronese variety V n,d, then the

hypersurface ν−1n,d(H) has a singular point in ν−1

n,d(P ). From this fact, the Sec-

ond Terracini’s Lemma in the case of Veronese varieties can be rephrased as

follows.

Corollary 2.14. Every hypersurface of degree d in Pn which is singular at a

general set of points P1, . . . , Ps is also singular along a positive dimensional

variety C passing through P1, . . . , Ps.

Using this fact, we are able to prove the Alexander-Hirschowitz Theorem

in the case of P2. We can remark that the case of conics is been already seen in

the previous sections from many point of view, in fact it is the case of Veronese

surface embedded in P5 which has the second secant variety defective, i.e.,

in our new language, a set of 2 generic double points of P2 does not impose

independent condition OP2(2) (see Example 1.8, Example 2.1 and Remark 2.3).

Theorem 2.15. A general family of double points X ⊂ P2 imposes indepen-

dent conditions on the plane curves of degree d with the only exceptions

� d = 2, s = 2;

� d = 4, s = 5.

Proof. We can easily check the statement for small values of d.

� d = 1: it is trivial. There are not curves of degree 1 with singular points

which is exactly what we expected;

� d = 2: let X be the set of two double points. The double line passing

through such points is also singular at these. Thus, the dimension of

IX(2) is at least 1 and then the codimension is

codim IX(2) ≤(

2 + 2

2

)− 1 = 5 < min

{(2 + 2

2

), 3 · 2

}= 6.


Moreover, since X does not impose independent condition on OP2(2), by

Lemma 3.2 such conic is also unique and the codimension of IX(2) is

actually 5.

Now, if s = 1 we may assume that P = [1 : 0 : 0]. Imposing the passage

to P and the vanishing of any partial derivatives at P , we obtain that

such conics need to be of the kind a11x21 + a12x1x2 + a22x

22. So

codim I2P (2) =

(2 + 2

2

)− 3 = 3 = min

{(2 + 2

2

), 3 · 1

}.

If s ≥ 3, since there is only one conic with 2 double points, just imposing

the passage through a third general point we lose such curve. Hence the

codimension is actually(

2+22

)as we expect.

� d = 3: every cubic with two double points contains the line through these

two points by the Bezout Theorem, hence the unique cubic with three

double points is the union of three lines passing pairwise through such

points. Like above, the dimension of IX(3) is 1, hence

codim IX(3) =

(2 + 3

2

)− 1 = 9 = min

{(2 + 3

2

), 3 · 3

}.

For s < 3, we have also that X does impose independent conditions on

OP2(3) by Remark 2.6. Instead, if s > 3, since there is only cubic with

three singular point, it is enough to impose to passage through a fourth

point to lost this conic from our ideal.

� d = 4: Let X be the set of five double points in general position. The

double conic passing through such points is also singular at these points,

so the dimension of IX(4) is at least 1 and hence

codim IX(4) ≤(

2 + 4

2

)− 1 = 14 < min

{(2 + 4

2

), 3 · 5

}= 15.

Moreover by Lemma 3.2 it is unique, thus codim IX(4) is exactly 14.

So, for s > 5, like the cases above, we get that the dimension of IX(4)

becomes 0 and we conclude that sets of s > 5 double points in general


position impose independent conditions on OP2(4). Instead, for s ≤ 4, we

get that they impose independent conditions by the Remark 2.7. Indeed,

we can find a cubic singular at three of them but which doesn’t pass

through the fourth, and also a cubic which is singular to three points and

has any possible tangency at the fourth. In the following figures we try

to describe this fact.

Figure 2.1: 4 general double points impose independent conditions on cubics.

Now, we want to prove that in degree bigger than 4 we have not defective

cases. Assume that X is a general union of s double points which does not

impose independent conditions on the plane curves of degree d. Let F be a a

plane curve through X of degree d, then, by Lemma 3.2, F contains a double

curve of degree 2l through X. Hence, we get the inequalities

2l ≤ d and s ≤ l(l + 3)

2=

(l + 2

2

)− 1.

We may also assume ⌊1

3

(d+ 2

2

)⌋≤ s

because the left-hand side is the maximum expected number of double points

imposing independent conditions on plane curves of degree d, so that we get⌊(d+ 2)(d+ 1)

6

⌋≤ d

4

(d

2+ 3

)


which gives d ≤ 4 (already considered) or d = 6. So that, we need only to

consider this last case: the last inequality is an equality and it forces s = 9.

Now, by Lemma 3.2, we can see the unique sextic with nine double points is

the double cubic, hence the dimension of IX(6) is 1 and

codim IX(6) =

(2 + 6

6

)− 1 = 27 = min

{(2 + 6

2

), 3 · 9

}.

Remark 2.8. Before going further to the study of exceptional cases, we want

to stress the fact that the case of P2 is very special. In fact, the proof of the

Alexander-Hirschowitz Theorem in P2 uses simple argumentations based above

all on the Second Terracini’s Lemma.

Increasing the dimension of the ambient space, we cannot hope that the

same argumentations give the same strong information on the nature of hy-

persurfaces of a given degree and with certain singularities. The “positive

dimensional variety” given by Lemma 3.2 gives us, in the case of P2, a lot of

information because it needs to be a curve and then an hypersurface in P2.

Increasing the dimension of the ambient space, the codimension of such vari-

ety could be too big to give us sufficient information to attack the cases with

larger dimension using similar argumentation. Thus we need to use completely

different methods.

Example 2.4. Let’s to consider the case of cubics in P3. The dimension of

OP3(3) is(

3+33

)= 20, hence we expect that there is not cubics in P3 with 5

double points.

We could prove that it is actually true and hence a set of 5 double points

impose independent conditions on cubics of P3 just by using “brute force”. The

idea is to assume that such points are the five fundamental points in P3 and

then apply the conditions on the generic cubic.

Fortunately, we can also use a faster and smarter geometrical method.

By Remark 2.7, using the same idea described in Figure 2.1, we can say

that 4 double points impose independent conditions on cubics and than the


dimension of the linear system L of cubics with 4 singular points is equal to 4.

By assuming that such points are the points P0 = [1 : 0 : 0 : 0], . . . , P3 = [0 :

0 : 0 : 1], we get a basis for L considering

{x1x2x3, x0x2x3, x0x1x3, x0x1x2},

i.e. the four possible triple of lines connecting three points between P0, . . . , P3.

Now, we can consider the Cremona transformation given by such lynear system

P3 99K P3

[x0 : x1 : x2 : x3] 7−→ [x1x2x3 : x0x2x3 : x0x1x3 : x0x1x2]

which is a birational map. By this transformation, we have that the hyper-

surfaces of degree 3 and singular at P0, . . . , P3 are mapped to an hyperplane

of P3. If we admit that one of such hypersurfaces has a fifth singular point,

then, after the Cremona transformation, we would have an hyperplane with a

singularity which is obviously absurd.

Thus, we have no cubics in P3 with 5 singular points and then they impose

independent conditions on OP3(3).

Remark 2.9 (Weakly defectiveness). The Second Terracini’s Lemma 3.2

suggests the following definition.

Definition 2.4. A variety V in PN is said to be s-weakly defective if the general

hyperplane H ⊂ PN tangent at s+ 1 general points of X, is also tangent along

a positive dimensional subvariety passing through the tangent points.

By Lemma 3.2 we have that

if a variety is s-defective, then it is also s-weakly defective.

The converse is not true in general and we have just seen a very easy

counterexample, the Veronese surface V 2,6 in P27. In the last part of the proof

of Theorem 2.15, we have seen that sets of double points impose independent

conditions on the sextics on P2, or equivalently that the surface V 2,6 is never

defective. However, we have seen that if we take 9 points in general position,


there exists a sextic singular at such points which is the double cubic. Of

course, since it is a double curve, it is also singular at every points.

This fact impose that the hyperplane in P27 given by the Veronese embed-

ding ν2,6 of such sextic which is tangent to the Veronese surface V 2,6 along an

entire curve. Consequently, V 2,6 is 9-weakly defective but non 9-defective.

The study of weakly defectiveness is useful to study the nature of defective

varieties, but in this thesis we won’t develop more this notions. A reference for

this kind of problem is [CC02].

2.3.2 The exceptional cases

As we have already said, the first, and easier, step to move toward the prove

of the Alexander-Hirschowitz Theorem is to understand the exceptional cases.

In the previous section we have seen the case n = 2, now we prove the cases

n > 2.

Remark 2.10 (d = 2, 2 ≤ s ≤ n). Set of k general 2-fat points does

not impose independent conditions on OPn(2).

In order to understand the spirit of the problem, let k = 2. Without loss

of generality, we may assume that the support of the two double points is

X = {P = [1 : 0 : . . . : 0] and Q = [0 : 1 : 0 : . . . : 0]} .

Consider the generic quadric in Pn

F = a00x20 + a01x0x1 + . . .+ annx

2n,

we impose that the corresponding hypersurface pass through P and Q and it

is singular at such points:

· passage through P implies a00 = 0;

· passage through Q implies a11 = 0;

· singularity at P implies a0i = 0 for each i = 1, . . . , n;


· singularity at Q implies a1i = 0 for each i = 2, . . . , n.

In this way, we get that F becomes a cone with vertex which contains the

line through the points P and Q and then the space of such hypersurfaces has

dimension(n2

)which is always greater than

(n+2

2

)− 2(n+ 1). Thus

codim IX(2) =

(n+ 2

2

)−(n

2

)< 2(n+ 1) ≤ min

{(n+ 2

2

), 2(n+ 1)

}.

The same argument works for k general points, thus we get that any family

of k 2-fat points in Pn, with 2 ≤ k ≤ n, does not impose independent condition

to the linear system of quadrics.

Remark 2.11 (d = 4, n = 3, s = 9 and d = 4, n = 4, s = 14). 9 and 14

2-fat points does not impose independent conditions on OPn(4) re-

spectively for n = 3, 4.

Through s =(n+2

2

)− 1 generic points in Pn, we have a unique and smooth

quadric and hence we get a quartic which is singular at our set of points,

simply we have to consider this double quadric. Meanwhile, we have that(n+4

4

)≤ (n + 1)

[(n+2

2

)− 1]

exactly for 2 ≤ n ≤ 4 and, in our numerical

hypothesis, we obtain the inequality:

codim IX(4) =

(n+ 4

4

)− 1 <

(n+ 4

4

)= min

{(n+ 2

2

), 2(n+ 1)

}.

It remains the only to consider the hypersurfaces of odd degree: this numer-

ical difference does not allow us to use an argument similar to previous cases.

Before we have to recall, without proof, some basic fact about the Rational

Normal Curves, i.e. the Veronese embedding of P1 in Pd.

One of the most important and useful property of the Rational Normal

Curve Cd in Pd is that any d+ 1 points on Cd are independents: it is equivalent

to the fact that Van der Monde determinant vanishes if and only if two rows are

proportionals. This is one of the key point to prove a lot of relevant features

about the Rational Normal Curve.

One of those, crucial in our following claims, is a very classical result:


Theorem 2.16 (Castelnuovo). (In [Har77], Theorem 1.18.) For any d+ 3

points in Pd in general position there exists a unique Rational Normal Curve.

A very useful way to describe the Rational Normal Curve in Pd is as the

zero locus of 2× 2 minors of the matrix

x0 x1 x2 . . . xh

x1 x2 . . . . xh+1

x2 . . . . . ....

......

......

. . . . . . xd−1

xd−h . . . . . xd

,

this is an example of the so-called determinantal varieties. Moreover, we can

generalize this determinantal representation to the secant varieties of the Ra-

tional Normal Curve:

Proposition 2.17. (In [Har77], Theorem 9.7.) For any s ≤ h, d − h, the

s-secant variety of the Rational Normal Curve Cd is the zero locus of the s× s

minors of the matrix

x0 x1 x2 . . . xd−h

x1 x2 . . . . xd−h+1

x2 . . . . . ....

......

......

. . . . . . xd−1

xh xh+1 . . . . xd

.

With this properties of the Rational Normal Curve, we are able to prove

the last exceptional case:

Remark 2.12 (d = 3, n = 4, s = 7). 7 2-fat points in P4 does not

impose independent conditions on OP4(3).

Since we get that(

4+33

)= 7 · 5, we expect that there are no cubics in P4

with 7 singular points, but by the Castelnuovo’s result stated above, through


seven points in P4 there is a unique Rational Normal Curve C4 which is the

zero locus of the 2× 2 minors of the matrixx0 x1 x2

x1 x2 x3

x2 x3 x4

.Moreover, its 2-secant variety has equation

det

x0 x1 x2

x1 x2 x3

x2 x3 x4

= 0.

So that, σ2 (C4) is a hypersurface on P4 of degree 3 singular along the whole

curve C4.

In this way, we have proved the defectiveness of all exceptional cases. In the

following section we will explain the methods used to cover the non-defective

cases.

2.3.3 Terracini’s inductive method

Terracini studied the dimension of the linear system of hypersurfaces passing

through a subscheme of double points of P3, by specializing some of them on

a plane P2 ⊂ P3. This is a basic example of a very useful inductive method

which covers many cases using a simple argument, but unfortunately, it will be

not enough to prove all the instances. In order to cover all cases, we will need a

degeneration method which is been the main contribution given by Alexander

and Hirschowitz.

Let X = 2P1 + . . . + 2Ps be a set of double points in Pn with ideal IX =

℘21 ∩ . . . ∩ ℘2

s, and let H be an hyperplane in Pn defined by a linear form F .

Suppose P1, . . . , Pr lying on H. Developing the original Terracini’s idea, we

can consider two schemes:


the trace X ∩H and the residue X.

The intersection X∩H is the subscheme of Pn associated to the saturation of

the ideal IX+(F ), but it can also be viewed as the subscheme of the hyperplane

H ' Pn−1 defined by the ideal IX∩H = IH(P1)2 ∩ . . .∩ IH(Pr)2 where IH(Pi) ⊂

k[x′0, . . . , x′n−1] ' k[x0, . . . , xn]/(F ) is the prime ideal associated to the point

Pi viewed on the hyperplane H.

The residue X geometrically is the subscheme of Pn which we get by cutting

off the hyperplane H, i.e. the scheme P1 + . . . + Pr + 2Pr+1 + . . . + 2Ps of r

simple points lying on a plane and s− r double points. Algebraically speaking

X is the scheme associated to the ideal IX : (F ).

Example 2.5. Consider the set of two double points in P2:

X = {P0 = [1 : 0 : 0] and P1 = [0 : 1 : 0]} ,

hence the ideal of this 0-dimensional scheme will be

IX = ℘20 ∩ ℘2

1 = (y, z)2 ∩ (x, z)2 = (xyz, xy2z, x2y2, x2yz, z2) ⊂ k[x, y, z].

Now, let L be the line passing through P0 and P1, represented by the ideal

IL = (z) ⊂ k[x, y, z].


After that, we consider the intersection between the two schemes, X∩L which

is represented by the sum of two ideals

IX∩L = (xyz, xy2z, x2y2, x2yz, z2) + (z) = (x2y2, z),

and the residue X with the ideal

IX = IX : IL = (z, xy).

Geometrically, we can easily understand what this ideals mean:

IX = (xy, z) = (x, z) ∩ (y, z)

represents the 0-dimensional scheme in P2 of the simple points P0 and P1;

instead, the intersection

IX∩L = (z, x2y2)

viewed in the quotient k[x, y, z]/(z) ' k[x, y] which is the coordinate ring of

the line L, it becomes (x2y2) = (x)2 ∩ (y)2 so that it represents two double

point on the line L ' P1.

Now, from the above definitions, we get an inclusion of k-vector spaces

[IX]d−1 ↪→ [IX]d given by the multiplication by F , moreover the saturation of

the cokernel is the dth-piece of the ideal of the trace of X on H, so that we get

the exact sequence

0 −→ [IX]d−1 −→ [IX]d −→ [IX∩H ]d, (2.6)

which is usually called the Castelnuovo’s exact sequence.

This is the core of the inductive argument that we are going to use. Since

the sequence above is exact we get the following inequality

dimk[IX]d ≤ dimk[IX]d−1 + dimk[IX∩H ]d. (2.7)

Since always holds the following property about binomial coefficients(n+ d

n

)=

(n+ d− 1

n

)+

(n+ d− 1

n− 1

),


we can rephrase this inequality in terms of Hilbert function

HFX(d) ≥ HFX(d− 1) + HFX∩H(d).

Hence, since we always have the lower bound

dimk[IX]d ≥ max

{0,

(n+ d

d

)− s(n+ 1)

}, (2.8)

we will look for some numerical conditions such that the two bounds become

equals in order to compute the dimension of [IX]d.

We say that it is an inductive method because looking at the exact sequence

the right hand side involved hypersurfaces of degree d − 1 and the left hand

side involved hypersurfaces in Pn−1.

Theorem 2.18. Let X be a union of s double points of Pn with support at

{P1, . . . , Ps} and fix H an hyperplane containing exactly r of them. Assume

that the scheme X∩H = 2P1|H+. . .+2Pr|H does impose independent conditions

on OPn−1(d) and the residual X = P1 + . . .+Pr + 2Pr+1 + . . .+ 2Ps does impose

independent condition on OPn(d − 1). Moreover, assume one of the following

arithmetical conditions:

(i) rn ≤(d+n−1n−1

)& s(n+ 1)− rn ≤

(d+n−1n

);

(ii) rn ≥(d+n−1n−1

)& s(n+ 1)− rn ≥

(d+n−1n

);

Then X does impose independent conditions on OPn(d).

Proof. By considering the exact sequence 2.6, we get the inequality

dimk[IX]t ≤ dimk[IX]t−1 + dimk[IX∩H ]d.

Since X ∩H imposes independent conditions on OPn−1(d), we know that

dimk[IX∩H ]d = max

{0,

(d+ n− 1

n− 1

)− rn

};

similarly, since X imposes independent conditions on OPn(d− 1), we also have

dimk[IX]d−1 = max

{0,

(d− 1 + n

n

)− (s− r)(n+ 1)− r

}.


Now, in the case (i) we get

dimk[IX]d ≤(d+ n− 1

n− 1

)− rn+

(d− 1 + n

n

)− (s− r)(n+ 1)− r =

=

(d+ n

n

)− s(n+ 1).

In the case (ii), we get dimk[IX]d ≤ 0.

Since always holds the inequality 2.8, we conclude.

Remark 2.13. The theorem above, which will be crucial to cover a lot of non

exceptional cases, talk about families of points with some of those lying on a

hyperplane, hence a natural question arises: since the Alexander-Hirschowitz

Theorem talks about sets of points in general position,

how can Theorem 2.18 be useful to our problem?

The idea is the following. If we want to prove that a scheme X of s dou-

ble points imposes independent condition on OPn(d), then we consider a new

scheme Y obtained by specializing some of them on an hyperplane. By the

semicontinuity of the Hilbert function, we get that

HFY(d) ≤ HFX(d).

Since we always have the upper bound for the Hilbert function of a scheme of

s double points in Pn which is

HFX(d) ≤ min

{s(n+ 1),

(n+ d

n

)},

if we prove that the scheme Y of s double points after the specialization imposes

independent conditions on OPn(d), a fortiori, also the scheme X of s double

points in general position imposes independent conditions.

In other words, given a set of s double points, we will try to specialize some

of them on an hyperplane so that one of the conditions of Theorem 2.18 will

be satisfied.

Example 2.6. Let us see some examples in P3.


We have already proved that the general double points impose independent

conditions on cubics in Example 2.4. Now, we are going to increase the degree

of the linear system OP3(d) in order to prove the non-defective case. The case

of cubics in P3 and of plane curves given by Theorem 2.15 will be the “base

cases” for this sort of inductive method.

Consider the case of quartics, d = 4. We have just proved that 9 general

double points does not impose independent condition, so that consider the

case of a scheme X of 8 general double points. The degree of this scheme is

8 · 4 = 32 and since OP3(4) has dimension(

4+33

)= 35, so that we expect that

the dimension of IX(4) is equal to 3.

Let us specialize 4 points on an hyperplane H in a such way they are in

general position on H. We can easily check that the inequalities of case (i) of

Theorem 2.18 are satisfied:

4 · 3 = 12 ≤ 15 =

(4 + 3− 1

3− 1

);

8(3 + 1)− 4 · 3 = 20 ≤ 20 =

(4 + 3− 1

3

).

Now, we know that, by Theorem 2.15, 4 double points in general position

impose independent conditions on OP2(4). Hence,

dim IX∩H(4) = max

{0,

(2 + 4

2

)− 4 · 3

}= 3.

Moreover, the set of 4 double points in general position and 4 simple points on

a plane imposes independent condition on OP3(3). The expected dimension for

the ideal of the the residue in degree 3 is

expdim IX(3) = max

{0,

(3 + 3

3

)− 4 · 4− 4

}= 0.

Claim: there is no cubics in P3 passing through 4 given points in general

position on a plane H and 4 double points away from such plane.

Indeed, since we have proved in Example 2.4 that the case of cubics in P3 is

never defective, 4 double points in P3 impose independent condition on OP3(3),


so that the dimension of the set of cubic surfaces singular at 4 general points

is equal to 4. Since there doesn’t exist cubic which are the union of a quadric

through 4 double points and a plane, after imposing the passage through the

4 simple points which lie on H, the dimension of such set of quartics becomes

0 and the claim is proved.

Now, we are able to apply the Theorem 2.18 and conclude that 8 double

points in general position impose independent condition on OP3(4).

We also can check numerically that the statement works. As we have said

the expected dimension of 8 double points in P3in degree 4 is 3, so that we

have the trivial upper bound

dim IX(4) ≤ 3;

by using the Castelnuovo’s exact sequence, we get also the lower bound

that we need to conclude:

dim IX(3) + dim IX∩H(4) = 0 + 3 ≤ dim IX(4) ≤ 3.

By Remark 2.6, we can also conclude that any set of s ≤ 8 double points

in general position impose independent condition on OP3(4).

Consider now the case d = 5. The dimension of OP3(5) is equal to(

3+53

)=

56. Since in P3 a double point has degree 4, if we consider a family of s = 14

double points, the degree is equal to the dimension of OP3(5) so that if they

impose independent conditions, by Remark 2.6, any set of double points does

the same. In order to understand how many points we have to specialize to

try to conclude, we can see that

the dimension of OP3(4) =(

3+43

)= 35;

the dimension of OP2(5) =(

2+52

)= 21.

Thus, we specialize u = 7 points on an hyperplane H so that the degree of the

trace X ∩ H is 21 and the degree of the residue X is 4 · 7 + 7 = 35. We can


check that we are in the arithmetical hypothesis of Theorem 2.18, hence if we

prove that both of trace and residue impose independent conditions on OP2(5)

and OP3(4), respectively, we are done.

By our knowledge on the case of quintics in P2 and quartics in P3 that

we have already considered, we can say that 7 general double points impose

independent condition on OP2(5) and, on the other hand, also the residue

imposes independent conditions on OP3(4). In fact, it follows by using a similar

argument as above, since 7 double points imposes independent condition on

OP3(4) and there doesn’t exist a quartic which is union of a cubic singular at

7 points and a plane.

Again by Theorem 2.18, we get that a set of 14 double points impose

independent conditions on OP3(5), so that, by using Remark 2.6, a general set

of s double points impose independent conditions on the quintic surfaces of P3.

Unfortunately, we are soon forced to give up the easy inductive method

given by Theorem 2.18. In fact there are arithmetical cases for what both of

the inequalities which are in the hypothesis of the theorem cannot be satisfied.

The first example is precisely in degree d = 6 and for set of 21 general double

points in P3.

In order to solve these arithmetical problems, Alexander and Hirschowitz

have introduced a clever degeneration argument, the so-called “methode d’Horace

differentielle”.

2.3.4 ”La methode d’Horace differentielle”

In this last section about the Alexander-Hirschowitz Theorem, we want to give

an idea about the method introduced to solve the last arithmetical.

In order to give an idea about the “Horace differentielle” and how it works,

we begin considering an easy numerical case which we have already considered

in Example 2.4. The set of 5 double points in general position in P3 imposes

independent conditions on the linear system OP3(3). We want to prove it with


the Horace differential method.

What we expect is that there is no cubics with 5 singularities. If we want

to use the inductive method that we have described in the previous section,

we have to specialize some points of X on an plane H, but we can immediately

understand where is the limit of this standard argument.

We want to specialize some points so that the trace X∩H and the residue

X impose independent condition respectively on OP2(3) and OP3(2) which are

spaces of dimension 10.

If we specialize 3 points on the plane, the residue X is the scheme union of

three simple points and two double points, so that the degree is 11 which is

greater that the dimension of OP3(2) and we have one unnecessary condition.

If we specialize 4 points on the plane, the trace X∩H is a scheme in P2 union

of 4 double points, so that the degree is 4 · 3 which is greater than we need

and, again, we have some useless condition.

This arithmetical problem can be overcome by using the Horace differentielle.

In the standard inductive method in Pn, when we specialize a fat point of de-

gree n+ 1 on an hyperplane H, it counts as a double point of degree n in the

trace X∩H and as a simple point of degree 1 in the residue. Using the Horace

differentielle we get the converse, heuristically it counts as a simple point on

the trace and as a point of degree n in the residue.

In our example, it is enough to get the thesis to consider 3 points specialized

on an plane H, a point differentially specialized on the hyperplane and a double

point out of the hyperplane. Thus, the trace X ∩ H is the scheme union of

three double points and a simple point on H, and the residue X is the union

of three simple points, a double point on H, i.e. a point of degree 3 in P3, and

a double point. So,

deg(X ∩H) = 10 = deg(X),

as we needed. Now, if we prove that they impose independent conditions, i.e.

there is no plane cubics passing through the trace and no quadrics in P3 passing

through the residue, we are done.


Consider the residue X. The quadrics passing through a double points are

cones, so that if we impose the passage through other 3 simple point and a

double point which lie on the plane, we get a linear system of dimension equal

to the dimension of conics in P2 passing through a double point and 3 simple

points in general position. Immediately we obtain that such dimension is 0, so

that the residue X impose independent conditions on OP3(2).

Now, consider the trace X ∩ H. We are looking for a cubic in P2 passing

through three double points and a simple point in general position. Since a

cubic passing through three 2-fat points need to be the union of the three lines

which connect pairwise such points, there is no cubic passing through the trace.

So, X ∩H impose independent conditions on OP2(3).

In summary, we have proved again that 5 fat points impose independent

conditions on OP3(3), but using the idea of the Horace differentielle.

This is the heuristic idea behind this method introduced by Alexander and

Hirschowitz, but now we want to explain why it works.


Definition 2.5. A 0-dimensional scheme is called curvilinear if it is contained

in a non singular curve.

Immediately from the definition we get that a curvilinear scheme contained

in a union of s double points has degree smaller than or equal to 2s.

In order to give the following crucial lemma in the general setting, if D is a

linear system on Pn, we say that a 0-dimensional scheme X is D-independent

if the Hilbert function of X with respect D is equal to the degree of the scheme

X. Namely, for whom are confident with cohomology

hPn(X,D) := dim H0(D)− dim H0(IX ⊗D) = deg(X).

We may recall that if we consider the linear system D = OPn(d), then H0(IX⊗

D) = IX(d) and the Hilbert function is the usual

hPn(X,D) = HFX(d) =

(n+ d

n

)− dimkIX(d).

Lemma 2.19 (Curvilinear Lemma, Chandler). (In [Cha01], Lemma 4.)

Let X ⊂ Pn be a 0-dimensional scheme contained in a finite union of dou-

ble points and D be a linear system on Pn such that deg(X) ≤ dim H0(D).

Then X is D−independent if and only if every curvilinear subscheme of X is

D−independent.

This lemma is a generalization of Remark 2.7. In fact, let X a given 0-

dimensional scheme in Pn with degree less or equal to dim OPn(d) + n + 1.

Thus, we can consider the linear system D = OPn(d) ⊗ IX. If we want to

understand if one more double point 2P imposes independent conditions on D,

by Curvilinear Lemma, we just have to check if any curvilinear subscheme of

2P imposes independent conditions on D. Since a curvilinear subscheme of 2P

has at most degree 2, can only be the simple point or the simple point with a

direction this is exactly to check if both the passage through P and the possible

tangency at P are independent conditions on the hypersurfaces of degree d

passing through X. Moreover, Curvilinear Lemma says that this argument


can be generalized if we want to understand if the addition of a generic 0-

dimensional scheme, non necessarily one double point, impose independent

condition on a linear system D.

Now, we want to explain the rigorous application of the Horace differentielle

on the example of five double points on the linear system OP3(3).

The idea is to consider a flat family of

points {Pt}t∈k and hyperplane {Ht}t∈k,

i.e. a family of points parametrizes on

t ∈ k, such that Pt ∈ Ht and Pt /∈ H for

any t 6= 1 and P1 ∈ H = H1, P0 = D.

Now, it is enough to prove that there exists a point Pt such that the scheme

2Pt imposes independent conditions on OP3(3) with 2A, 2B, 2C and 2E.

Assume by contradiction that it is false, by the Curvilinear Lemma, for

each point Pt of the flat family there exists a curvilinear subscheme δt of 2Pt,

namely a scheme of degree at most 2 with support in Pt, such that it doesn’t

impose independent conditions, i.e.

HF2A+2B+2C+δt+2E(3) < min

{(3 + 3

3

), 4 · 4 + 2

}= 18.

By the semicontinuity of the Hilbert function, i.e. the fact that when we

specialize a scheme the Hilbert function can only decrease, we get

HF2A+2B+2C+δ0+2E(3) ≤ HF2A+2B+2C+δt+2E(3) < 18. (2.9)

Now, we have two possibilities.

δ1 6⊂ H: we consider the scheme X = 2A+ 2B + 2C + δ1 + 2E and we use the

Castelnuovo’s exact sequence with respect the hyperplane H. Since δ1

has degree 2 with support on H, but it is not contained in H, the trace

is the union of three double point and a simple point which we have seen

that it imposes independent conditions on OP2(3), hence

dimkIX∩H(3) = max

{0,

(3 + 2

2

)− (3 · 3 + 1)

}= 0.


Instead, the residue is four simple points on a plane and a double point.

Since the quadrics through a fat point are cones, the space of quadrics

passing through the residue has the same dimension of the conics passing

through four general points. Since four simple points impose independent

conditions to conics, we get that

dimkIX(2) = max

{0,

(2 + 2

2

)− 4

}= 2.

In summary, we get that, in terms of Hilbert function, by using the

inequality given by the Castelnuovo’s exact sequence

HFX(3) ≥ HFX(2) + HFX∩H(3) = 10 + 8 = 18,

which is in contradiction with 2.9.

δ1 ⊂ H: by semicontinuity of the Hilbert function, we know that there exists

an open neighborhood U of 1 such that

HFA+B+C+2Pt|Ht+2E(2) ≥ HFA+B+C+2P1|H+2E(2) = 10,

and then the equality holds. So that, by the Curvilinear Lemma, also the

curvilinear subscheme δt impose independent conditions on OP3(2), i.e.

HFA+B+C+δt+2E(2) = 9.

Now, we take t ∈ U and apply again the Castelnuovo’s inequality to the

scheme X = 2A + 2B + 2C + δt + 2E. The trace on H is the scheme

X∩H = 2A+ 2B+ 2C which impose independent conditions on OP2(3),

i.e.

HFX(3) = min

{3 · 3,

(3 + 2

2

)}= 9.

By the Castelnuovo’s inequality,

HFX(3) ≥ HFX(3) + HFX∩H(2) = 9 + 9 = 18.

Again it is in contradiction with 2.9.


This complete the proof of the statement about the conditions imposed by

five 2-fat points on the cubics in P3 by using “la method d’Horace differentielle”.

In conclusion to this section about the Alexander-Hirschowitz Theorem, in

order to take more confidence with this kind of argumentation, we want to

apply the method to the first example for which the Horace differentielle is

actually necessary to conclude.

Proposition 2.20. The scheme X = {2A1, . . . , 2A21} of 21 general 2-fat points

on P3 impose independent conditions on the sextics OP3(6)

Proof. Since the degree of X is 21 · 4 = 84 and the dimension of OP3(6) is(3+6

3

)= 84, we expect that there is no sextics with 21 singular points.

Since the dimension of OP2(6) is(

6+22

)= 28, we can specialize on an hy-

perplane H at most 9 points. Let Φ = {2A1, . . . , 2A9} such points on an

hyperplane H, the degree of Φ|H is 9 · 3 = 27, so that we need to specialize

differentially one more point.

As before we consider the flat family of points {Pt}t∈k and hyperplane

{Ht}t∈k such that P0 = A10 and P1 ∈ H. Let Σ = {2A11, . . . , 2A21}. In order

to conclude the thesis, it is enough to prove that there exists a scheme 2Ptwhich impose independent conditions on the sextics with Σ and Φ.

Assume by contradiction that it is not true. By the Curvilinear Lemma,

for any t ∈ k there exists a curvilinear scheme δt supported on Pt with degree

at most 2, such that

HFΦ+δt+Σ(6) < deg(Φ + δt + Σ) = 82. (2.10)

Consider now the two different possibilities:

δ1 6⊂ H: we apply the Castelnuovo’s exact sequence to Y = Φ + δ1 + Σ. The

residue Y is 9 simple points from Φ and one simple point from δ1 lie on a

plane, and 11 2-fat points from Σ. Since a collection of 12 2-fat points in

P3 impose independent conditions on the quintics and there is no quartic


with 11 singular points and by Curvilinear lemma, the scheme Y impose

independent conditions and

HFY(5) = 54.

The trace Y∩H on H is the set of 9 2-fat points from Φ and one simple

point from δ1. Since 9 points in P2 impose independent conditions on

OP2(6),

HFY∩H(6) = 28.

In summary, by the Castelnuovo’s inequality

HFY(6) ≥ HFY(6) + HFY∩H(5) = 54 + 28 = 82,

in contradiction with 2.10.

δ1 ⊂ H by semicontinuity of the Hilbert function, there exists an open neigh-

borhood U of 1 such that

HFA0+...+A9+2Pt|Ht+Σ(5) ≥ HFA0+...+A9+2P1|H1+Σ(5) = 9 + 3 + 44 = 56,

and the equality holds. In particular, by Curvilinear Lemma, also replac-

ing Pt with a curvilinear subscheme δt impose independent conditions on

OP3(2), i.e.

HFA0+...+A9+δt+Σ(5) = 55.

Now, we take t ∈ U and apply the Castelnuovo’s inequalityto the scheme

Y = Φ + δt + Σ.

The trace on H is the scheme Y ∩ H = Φ which impose independent

conditions on OP2(6), i.e.

HFY∩H(6) = min

{3 · 9,

(2 + 6

2

)}= 27.

Since the residue is exactly Y = A0+. . .+A9+δt+Σ, by the Castelnuovo’s

inequality we get

HFY(6) ≥ HFY(6) + HFY∩H(5) = 55 + 27 = 82,

and again it is in contradiction with 2.10.


This conclude the proof.

Remark 2.14. In these examples we have differentially specialized only one

points, but in general one could have to use the differential method on many

points, but the main idea is exactly the same. The general statement of the

Horace differentielle is the following:

Theorem 2.21 (La methode de Horace differentielle). Let H ⊂ Pn

be a hyperplane, P1, . . . , Pr generic points in Pn and Z be a 0-dimensional

scheme. Let Z = Z + 2P1 + . . . + 2Pr ⊂ Pn, Z ′ the residue of Z with respect

toH and T the trace on H. Let P ′1, . . . , P′r be generic points in H and let

D2,H = 2P ′i ∩H and Z ′ = Z ′+D2,H(P ′1)+ . . .+D2,H(P ′r), T = T +P ′1 + . . .+P ′r.

If

dim IZ′(t− 1) = 0 and

dim IT (t) = 0,

then dim IZ(t) = 0.

2.4 Open problems

In this Chapter we have talked about the defectiveness of secant varieties of

Veronese varieties: as we have already largely said this is a very special case

because is the unique family of projective varieties for which we have a complete

characterization in terms of the dimension of secants.

Nevertheless, also for Veronese varieties there are many open problems and

the secant varieties are largely unknown: in general, we don’t know the equa-

tions or the minimal free resolution. We have rather complete informations

about the commutative algebra associated to the Rational Normal Curve, i.e.

the Veronese embeddings of P1, and the quadratic Veronese embedding. In

these cases we know explicitly the equations and the minimal free resolutions,

but they are very special cases because those are determinantal varieties, so

that the computation is easier.

2.4 Open problems 2. Veronese varieties

This lack of knowledge is reflected on the applications, namely on the War-

ing problem for polynomials. The Alexander-Hirschowitz Theorem gives a

solution of the “Big” Waring problem, i.e. to find the rank of the generic form

of a given degree, but to solve the “Little” Waring problem, i.e. to find the

rank of each form of a given degree, we need a large knowledge of the secant

varieties of Veronese varieties and this is still unsolved in general.

Up to know, we have an algorithm to compute the rank of binary forms,

which is the case connected to the secants of the Rational Normal Curve and it

is classically attributed to Sylvester, but it can be viewed in term of apolarity

that we have described previously. The key point is the “Apolarity Lemma”

that we can state as follows:

Lemma 2.22 (Apolarity Lemma). Let F ∈ Sd be a form of degree d in

n+ 1 variables. Then the following are equivalent:

(i) F = Ld1 + . . .+ Lds;

(ii) F⊥ ⊃ I such that I is the ideal of a set of s distinct points in Pn.

Moreover, we know the rank in the case of quadratic form and in a paper

of 2011 written by Carlini, Catalisano and Geramita [CCG11], they solved the

monomial case by using apolarity, in particular the main result was

rank(xa00 · . . . · xann ) =1

a0 + 1

n∏i=0

(ai + 1),

where 1 ≤ a0 ≤ . . . ≤ an.

So, the problem of secants of Veronese varieties could be very interesting

also after the Alexander-Hirschowitz Theorem and it could be a very fruitful

research field.

Chapter 3

Segre varieties

In this chapter we are going to talk about the defectiveness of another classical

and interesting family of projective varieties: the Segre varieties.

Unfortunately, we have not a complete description like the Alexander-

Hirschowitz Theorem, so that this case in largely still unsolved and we will

try to describe some specific recent results and the methods used.

We will refer to several papers of Catalisano, Geramita and Gimigliano:

the key point of their method once again the Terracini’s Lemma which will

convert the geometric problem of defectiveness to questions concerning the

Hilbert function of “fat points” in multiprojective spaces.

3.1 Tensor decomposition and “tensor rank”

As Veronese varieties are associated to the Waring’s problem for polynomi-

als since they parametrize the dth-powers of linear forms, the Segre varieties

will be connected to the “tensor decomposition” since they parametrize the

“decomposable tensors”.

In order to well understand these definitions, we will introduce the Segre

varieties in setting of multilinear algebra introduced in the first chapter.

Let V1, . . . , Vt be k-vector space of dimension respectively n1 +1, . . . , nt+1

97

3.1 Tensor decomposition and “tensor rank” 3. Segre varieties

over an algebraically closed field. Of course, we may assume n1 ≤ . . . ≤ nt.

Let B∗i = {xi,0, . . . , xi,ni} a basis for each V ∗i , so that

B∗ = {x1,j1 ⊗ . . .⊗ xt,jt | 0 ≤ ji ≤ ni, for i = 1, . . . , t}

is a basis for V = V ∗1 ⊗ . . .⊗V ∗t , in other words we can write each tensor T ∈ V

in the form

T =∑

0≤ji≤ni1≤i≤t

aj1,...,jtx1,j1 ⊗ . . .⊗ xt,jt =

=∑

0≤ji≤ni1≤i≤t−1

x1,j1 ⊗ . . .⊗ xt−1,jt−1 ⊗ yj1,...,jt−1 ,

where yj1,...,jt−1 =∑nt

jt=0 aj1,...,jtxt,jt .

Definition 3.1. A tensor T ∈ V is said to be decomposable if we can write

T = v∗1 ⊗ . . .⊗ v∗t , with v∗i ∈ V ∗i .

The minimal length of a representation of T as sum of decomposable is

called the tensor rank of T.

From the previous description of a generic tensor T as a sum, it is clear

that all tensors of V can be written as a sum of decomposable. Therefore, a

natural question arises:

(i) what is the least integer D(V) such that every tensor in V has tensor rank

less than D(V)?

(ii) what is the least integer E(V) such that the generic tensor in V has

tensor rank less than E(V)?

Clearly, these questions are similar to the Waring’s problem.

Let T ∈ V, by definitions it corresponds to a multilinear application

T : V1 × . . .× Vt → k.

If we take the basis Bi = {x∗i,0, . . . , x∗i,ni} on each Vi, say the dual basis to the

B∗i , T is completely determined by the values

T (x∗1,j1 , . . . , x∗t,jt) = aj1,...,jt .

3. Segre varieties 99

This fact justifies the common use to think a tensor of V as an “hypermatrix”

of size (n1 + 1) × . . . × (nt + 1). Hence, the tensors of V are parametrized,

up to multiplication by a scalar, by points in a projective space PN where

N =∏

(ni + 1)− 1. Now, we introduce the following polynomial rings:

Si = k[xi,0, . . . , xi,ni ] for all i = 1, . . . , t and

A = k[x1,0, . . . , x1,n1 ; . . . ;xt,0, . . . , xt,nt ].

We use on each Si the standard N-gradation and the consequent Nt-gradation

on A. It is clear that each k-vector space V ∗i can be identified with Si1 and V

with A1 where 1 = (1, . . . , 1).

From this point of view, we can define the Segre variety Seg(Pn1× . . .×Pnt)

as the image of the Segre embedding

ν1 : Pn1 × . . .× Pnt = PS11 × . . .× St1 → PA1

defined by

ν1(L1, . . . , Ls) = L1 ⊗ . . .⊗ Lt.

In this way, we get that the Segre product Seg(Pn1×. . .×Pnt) parametrized

the decomposable tensors in PN . Moreover, by definition of secant variety, the

generic tensor which lies on the s-secant variety has minimal decomposition in

decomposable tensors with length s, in other words, similarly to the Waring

problem, the computation of the dimension of secant varieties of a Segre prod-

ucts Seg(Pn1 × Pnt) allows us to know the tensor rank of the generic tensor of

size (n1, . . . , nt).

The computation of the tensor rank for a given tensor is quite harder and

needs a better knowledge of the nature of secant varieties of Segre varieties.

3.1.1. First case: two factors. Let n = (n1, n2) ∈ N2 and consider the

Segre product of two factors Pn1×Pn2 = P(V1)×P(V2). In order to understand

this case, consider the following classical description of the Segre embedding

3.1 Tensor decomposition and “tensor rank” 3. Segre varieties

ν1 : Pn1 × Pn2 −→ PN

([x0 : . . . : xn1 ], [y0 : . . . : yn2 ]) 7−→ [x0y0 : . . . : xiyj : . . . : xn1yn2 ],

which can also be written as a matrix productx0

...

xn1

· [y0 . . . yn1

]=

x0y0 . . . x0yn2

.... . .

...

xn1y0 . . . xn1yn2

.Thus, we may identify

V1 ⊗ V2 ↔ (n1 + 1)× (n2 + 1) matrices

decomposable vectors↔ matrices of rank ≤ 1.

From this point of view we get that Seg(Pn1 × Pn2) is the variety which

parametrizes the matrices of rank ≤ 1 and, since a matrix has rank ≤ s if and

only if is sum of ≤ s matrices of rank ≤ 1, the s-secant variety σs(Seg(Pn1 ×

Pn2)) is the space of matrices of rank ≤ s.

Example 3.1. Consider for example the Segre embedding of P2 × P2 into

P8. The space P8 can be viewed as the projective space of the generic 3 × 3

matrices. The ideal generated by the 2×2 minors of the generic matrix defines

the Segre variety Seg(P2 × P2) and the determinant gives the equation of the

secant variety σ2(Seg(P2 × P2)), so that the dimension is

dim σ2(Seg(P2 × P2)) = 7.

Instead, recalling that the dimension of the Segre variety is exactly what

one can expect, that is dim Seg(P2× P2) = 2 + 2 = 4, the expected dimension

of the secant variety, computed just by counting parameters is:

expdim σ2(Seg(P2 × P2)) = min{2 · 4 + 1, 8} = 8.

Hence, we have that this our first example of a defective Segre variety.

Actually, this happens for every case of Segre product with two factors,

except for P1 × P1, and it is easy to prove since all the theory is in term of


matrices. If we assume n1 ≤ n2, then for all s with 2 ≤ s ≤ n1 the secant

varieties σs(Seg(Pn1 × Pn2)) is always defective. Moreover, for s = n1 + 1

the secant variety fills the entire ambient space, in fact every matrix of size

(n1 +1)×(n2 +1) has rank ≤ n1 +1, or equivalently is sum of ≤ n1 +1 matrices

of rank 1.

We will see the proof of this fact as consequence of the first Terracini’s

lemma.

3.2 Yet again: Terracini’s Lemma

Similarly to the case of the Veronese varieties, we can use the Terracini’s lemma

to convert the problem of defectiveness of Segre varieties to the computation

of some Hilbert function. The difference from the Veronese case is that we will

work with multigradation on the polynomial ring.

We are going to find this relation in the basic case of two factors. The

generic case uses the same ideas, but it needs heavy notation. Consider the

Segre variety Seg(Pn×Pm) in P(n+1)(m+1)−1 = PN . we use notation introduced

previously, that is

S = k[x0, . . . , xn], R = k[y0, . . . , ym] and A = k[x0, . . . , xn; y0, . . . , ym],

so that Pn = P(S1), Pm = P(R1) and PN = P(A1).

Firstly we compute the tangent space at a point, we may assume P = ([1 :

0 : . . . : 0], [1 : 0 : . . . : 0]). As we have already done for the Veronese varieties

in the previous chapter, we compute the affine cone over the tangent space by

considering the line through our point with direction (N,M) ∈ S1 ×R1:

t 7→ (x0 + tN, y0 + tM),

so that the corresponding tangent vector in Tx0y0(A1) is given by

∂

∂t

∣∣∣P

(x0 + tN)(y0 + tM) =∂

∂t

∣∣∣P

(x0y0 + t(Ny0 + x0M) + t2NM) =

= Ny0 + x0M ∈ A1.

3.2 Yet again: Terracini’s Lemma 3. Segre varieties

In conclusion, we get that

Tx0y0(P(A1)) = 〈x0y0, x0y1, . . . , x0ym, x1y0, . . . , xny0〉.

Now, the key point is to observe that, given a hyperplane H ⊂ PN , then

ν−11 (H) can be viewed as a hypersurface of multidegree (1, 1) in the multipro-

jective space Pn×Pm, in fact the coordinates z0, . . . , zN in PN are the monomial

of multidegree (1, 1) in k[x; y].

If we consider our point P = ([1 : 0 : . . . : 0], [1 : 0 : . . . : 0]) and the tangent

space at it, we get that if an hyperplane H ⊂ PN contains Tx0y0(P(A1)), then

the hypersurface of multidegree (1, 1) ν−11 (H) needs to be

a1,1x1y1 + . . .+ a1,mx1ym + . . .+ an,1xny1 + . . .+ an,mxnym,

which pass through P and it is also singular at P .

By this fact, we get a bijection between

{hyperplane H ⊂ PN |H ⊃ TP1(P(A1)), . . . , TPs(P(A1)), for Pi ∈ Seg(Pn×Pm)}

and

{hypersurfaces of multidegree (1, 1) in Pn × Pm singular at P1, . . . , Ps} =

= [℘21 ∩ . . . ∩ ℘2

s](1,1),

where ℘ is the ideal of A defining the point Pi, e.g. if P = ([1 : 0 : . . . : 0], [1 :

0 : . . . : 0]), then ℘ = (x1, . . . , xn, y1, . . . , ym).

In summary, from Terracini’s Lemma and this bijection, we get

Lemma 3.1. Let Seg(Pn1 × . . . × Pnt) ⊂ PN , where N =∏t

i1(ni + 1) and

choose generic multiprojective points P1, . . . , Ps with defining ideals ℘1, . . . ℘s

respectively. Then

dim σs(Seg(Pn1 × . . .× Pnt)) = N − dimk[℘21 ∩ . . . ∩ ℘2

s](1,...,1) =

= HFA/℘21∩...∩℘2

s(1, . . . , 1).

So that, the problem of computing the dimension of secants of Segre vari-

eties is translated in a algebraic problem of computation of Hilbert functions,

but the multigradation makes trickier the problem.


Example 3.2. Using Terracini’s lemma, we are able to prove that, if n ≤ m,

the Segre product Seg(Pn × Pm) is always defective for s ≤ n. In fact we can

compute directly the dimension of the secant varieties.

Claim : dim σs(Seg(Pn×Pm)) = min{s(n+m+1)−s2+s−1, (n+1)(m+1)−1}.

Proof. Let Fbe a generic form corresponding to a generic point on the s-secant

variety σs(Seg(Pn × Pm)), we may assume that F =∑s

i=1NiMi, with Ni ∈ S1

and Mi ∈ R1. We may also suppose that the Ni’s and the Mi’s are independent.

By Terracini’s lemma and the computation that we have made above about

the tangent space of the Segre varieties, we get that

TP σs(Seg(Pn × Pm)) = 〈. . . , TNiMi(Seg(Pn × Pm)), . . .〉 =

= 〈. . . , S1Mi +NiR1, . . .〉 =

= S1〈M1, . . . ,Ms〉+ 〈N1, . . . , Ns〉R1.

Now, we can easily compute the intersection of these linear subspaces in

A(1,1) which is

S1〈M1, . . . ,Ms〉 ∩ 〈N1, . . . , Ns〉R1 = 〈N1, . . . , Ns〉〈M1, . . . ,Ms〉,

so that we have the following:

dimkS1〈M1, . . . ,Ms〉 = (n+ 1)s;

dimk〈N1, . . . , Ns〉R1 = s(m+ 1);

dimk〈N1, . . . , Ns〉〈M1, . . . ,Ms〉 = s2.

By the Grassmann’s formula, supposing that such tangent space doesn’t

feel the entire ambient space, the dimension of the s-secant variety is

dim σs(Seg(Pn × Pm)) = s(n+m+ 1)− s2 + s− 1.

Since the ambient space has dimension (n + 1)(m + 1) − 1 we get the

conclusion.

3.2 Yet again: Terracini’s Lemma 3. Segre varieties

Now, we can observe that for s ≤ n, the dimension of the s-secant variety

of Segre product Seg(Pn × Pm) is always

dim σs(Seg(Pn × Pm)) = s(n+m+ 1)− s2 + s− 1.

We also get

((n+ 1)(m+ 1)− 1)−(s(n+m+ 1)− s2 + s− 1) =

= (m+ 1)(n+ 1− s)− sn+ s2 − s =

= (m+ 1)(n+ 1− s)− s(n+ 1− s)

= (m+ 1− s)(n+ 1− s) > 0.

Since the expected dimension of this secant variety is

expdim σs(Seg(Pn × Pm)) = min{s(n+m) + s− 1, (n+ 1)(m+ 1)− 1},

it is clear that we have that, assuming n ≤ m,

for s ≤ n the s-secant varieties of Seg(Pn × Pm) is always defective.

Moreover, we can also observe that for s = n + 1, the dimension of the

s-secant variety of Seg(Pn× Pm) is equal to (n+ 1)(m+ 1)− 1 and hence it is

the first secant variety which fills the entire ambient space. In particular, it is

a non-defective case.

Remark 3.1. Using both of Veronese and Segre embedding, we can define an-

other very classical family of projective varieties, the so-called Segre-Veronese

varieties.

For any multiindex d = (d1, . . . , dt) ∈ Nt, we consider the following com-

position

Pn1 × . . .× Pnt // PN1 × . . .× PNt // PN ,

where the first map is given by the product of the Veronese embeddings of

degree di for i = 1, . . . , t, where Ni =(ni+dini

); the second map is the standard

Segre embedding, so that N = (N1 + 1) · · · . . . · · · (Nt + 1) − 1. The image of

such composition is the Segre-Veronese variety denoted with Vn,d.


Just to take more confidence with this definition, we explicit in components

the two factor case. The Segre-Veronese embedding of Pn1 × Pn2 in bi-degree

(d1, d2) ∈ N2 is

Pn1 × Pn2 −→ PN

([x0 : . . . : xn1 ], [y0 : . . . : yn2 ]) 7→ [xd10 yd20 : xd1−1

0 x1y0 : . . . : xd1n1yd2n2

].

Following the same argumentation for proving Lemma 3.1 in the case of

Segre varieties, we can prove that also for Segre-Veronese varieties in general

we can compute the dimension of secant varieties from a computation of Hilbert

function of 2-fat points in the multiprojective space Pn1 × . . .× Pnt .

dim(σs(Vn,d)) = HFA/℘21∩...∩℘2

t(d1, . . . , d2).

3.3 A combinatorial approach: the monomial

case.

Terracini’s Lemma gives us an interpretation of defectiveness of Segre varieties

in terms of multigraded Hilbert function of fat points. This multigradation

makes the computation harder, so that we have to find some trick to go further.

In the case of coordinate points, i.e. Pr = (Pr1 , . . . , Prt) ∈ Pn1 × . . .× Pnt ,

where Pri = [0 : . . . : 1 : . . . : 0] with 1 in rthi -position, the problem turns to a

monomial case which can be faced in a combinatorial way.

Definition 3.2. Let J = {r = ((r1, . . . , rt) | 0 ≤ ri ≤ ni}.

Given two vectors ri = (ri,1, . . . , ri,t) ∈ J, i = 1, 2, we define the Hamming

distance between r1 and r2 as∣∣{non-zero entries of the pointwise difference r1 − r2}∣∣.

Proposition 3.2. Let Pr1 , . . . , Prt be a set of coordinate points in Pn1 × . . .×

Pnt . If ℘i = (x1,0, . . . , x1,r1 , . . . , xi,ri , . . . , xt,nt) is the ideal of Pri , we consider

the scheme 2Pr1 + . . .+ 2Prt associated to the ideal⋂ti=1 ℘

2i . Then

HF2Pr1+...+2Prt(1, . . . , 1) =

∣∣∣ {r ∈ J∣∣ r has Hamming distance ≤1

from at least one of ri

} ∣∣∣.

3.3 A combinatorial approach: the monomial case. 3. Segre varieties

Proof. Firstly assume t = 1, i.e. our scheme is simply 2Pr, so that we have to

compute the Hilbert function of the monomial ideal

℘2 = (x1,0, . . . , x1,r1 , . . . , x1,n1 ; . . . ;xt,1, . . . , xt,rt , . . . , xt,nt)2.

So, since it is a monomial ideal, we are looking for the monomials of mul-

tidegree (1, . . . , 1) which are not in our ideal.

It is immediate to understand that xj /∈ ℘2 if and only if at most one entry

in j differs from r, or equivalently the Hamming distance between j and r is

less or equal to 1.

When t > 1, we know that xj /∈⋂ti=1 ℘

2i if and only if xj /∈ ℘2

i for some i,

so that the conclusion follows from the basic case.

This proposition allows us to visualize our problem as playing with “rooks

on a t-dimensional chessboard”. This is what we mean: let [n] = {0, . . . , n}

and define the chessboard [n] = [n1]× . . .× [nt], a coordinate point Pr may be

identified as rook on the board and a family of coordinate points becomes a

rook set.

Definition 3.3. Let R ⊂ [n] be a rook set. We define the subset generated by

R, denoted with 〈R〉, as the set of all the elements in [n] that can be obtained

by changing at most one coordinate of an element of R, i.e. the places which

can be “attacked” by rooks in R.

With this new language, we can rephrase the previous proposition as fol-

lows:


Proposition 3.3. Let Pr1 , . . . , Prt be a set of coordinate points in Pn1 × . . .×

Pnt . We consider R the rook set associated to our points and, if ℘i is the ideal

of Pri , the scheme 2Pr1 + . . . + 2Prt associated to the intersection⋂ti=1 ℘

2i .

Then,

HF2Pr1+...+2Prt(1, . . . , 1) = |〈R〉|

In summary, we have related problems dealing with secant varieties of Segre

product to rook set on a chessboard.

Definition 3.4. Let R be a rook set on [n]. It is said to be

(i) perfect if every element in 〈R〉 comes from exactly one element of R;

(ii) rook covering if 〈R〉 = [n];

(iii) perfect rook covering if both of the previous hold.

As a corollary of the Proposition 3.3, it immediately follows:

Corollary 3.4. Let R ⊂ [n] be a rook set with |R| = s. Then:

(i) if R is a rook covering, σs(Seg(Pn1 × . . .× Pnt)) = PN ;

(ii) if R is a perfect rook, dim σs′(Seg(Pn1×. . .×Pnt)) = s′(n1+. . .+nt+1)−1

for all s′ ≤ s;

(iii) if R is a perfect rook covering, then σs(Seg(Pn1 × . . . × Pnt)) = PN and

it is the first secant which fills the ambient space.

Using this point of view we can easily obtain the same result about the

product of two factors that we have proved in Example 3.2 by using Terracini’s

lemma, but also something about the product of more than three factors.

Proposition 3.5. Let Seg(Pn1 × . . . × Pnt) be the Segre product in PN with

n1 ≤ . . . ≤ nt. Then:

(i) if t = 2 and s = n1 + 1, then dim σs(Seg(Pn1 × Pn2)) = N ;

3.4 Multiprojective-affine-projective method 3. Segre varieties

(ii) if t = 2 and s ≤ n1, dim σs(Seg(Pn1 ×Pn2)) = s(n1 +n2 + 1)− s2 + s− 1;

(iii) if t ≥ 3 and s ≤ n1+1, dim σs(Seg(Pn1×. . .×Pnt)) = s(n1+. . .+nt+1)−1.

Proof. The proof of these facts needs only finding the right rook set.

(i) We can observe that if s = n1 + 1,

the main diagonal is always a trivial rook

covering with cardinality s, as in the fig-

ure.

(ii) We can use s places on the main diagonal. Then there are s(s − 1)

positions covered by two of the elements of R, so that R generates a set of

s(n1+n2+1)−s(s−1) elements. Actually, since such covering is not perfect and,

a priori, there could be a better choice for putting s rooks on the chessboard,

this proves only a lower bound for the dimension, i.e.

dim σs(Seg(Pn1 × Pn2)) ≥ s(n1 + n2 + 1)− s2 + s− 1.

However, since we are on a two dimensional chessboard, the coverings induced

by two rooks overlap at least 2 times and hence the coverings induced by s

rooks overlap at least 2 ·(s2

)= s(s − 1) times. In summary, the diagonal

configuration of rooks is not perfect, but the best possible, so that we get the

conclusion.

(iii) If t ≥ 3 and s ≤ n1 + 1, a perfect rook set of s elements can be

done taking s places on the main diagonal, so that once again we get the

conclusion.

Unfortunately, this combinatorial approach is useful to compute the Hilbert

function of multiprojective 2-fat points which can be reduced to coordinate

points. For larger values of s, we need to find other ways to attack the problem.

3.4 Multiprojective-affine-projective method

Using Terracini’s lemma, we have seen that the problem dealing with the de-

fectiveness of Segre varieties can be translate into a problem of computation of


multigraded Hilbert functions. The fact that we should work with multigrada-

tion makes hard the things.

In this paragraph we want to introduce without proof a very useful method

which allows to come back to Hilbert functions in the standard gradation.

This method can be called multiprojective-affine-projective method, and the

construction will make clear the reason for this strange name.

The aim of this method is to come back to standard gradation on the

polynomial ring, in other words to come back to a standard projective space

PN from the multiprojective space Pn1× . . .×Pnt . Clearly, the first expectation

is that N = n1+. . .+nt and it is actually the right thing, so let N = n1+. . .+nt.

First we consider the birational map

g : Pn1 × . . .Pnt //___ AN ,

defined by sending

([x1,0 : . . . : x1,n1 ]; . . . ; [xt,0 : . . . : xt,nt ])� //___

(x1,1x1,0

, . . . ,x1,n1x1,0

, . . . , xt,1xt,0

, . . . ,xt,ntx1,0

).

Such map is well-defined on the open subset of Pn1 × . . . × Pnt given by

{x1,0x2,0 . . . xt,0 6= 0}.

Now, if k[z0, . . . , zN ] is the coordinate ring of the projective space PN , we

can take simply the standard embedding of AN in the projective space PN

whose image is the affine chart {z0 6= 0}. By composing the map g with this

embedding, we get

f : Pn1 × . . .Pnt //___ PN ,defined by sending ([x1,0 : . . . : x1,n1 ]; . . . ; [xt,0 : . . . : xt,nt ])_

��

[1 : x1,1

x1,0: . . . :

x1,n1x1,0

: . . . : xt,1xt,0

: . . . :xt,ntx1,0

]=

= [x1,0x2,0 . . . xt,0 : x1,1x2,0 . . . xt,0 : . . . : x1,0 . . . xt−1,0xt,nt ].

If Z is a 0-dimensional subscheme in the multiprojective space Pn1×. . .×Pnt

contained in {x0,1 . . . x0,t 6= 0}, the idea is to construct a scheme in PN such


that the dimension in the standard graded ring in N + 1 variables [IW ]a is

exactly the dimension of [IZ ](a1,...,at) with a1 + . . . + at = a and we finally can

come back to the computation of Hilbert functions in the standard graded

polynomial ring.

Let us introduce some useful notation.

Let k[z0, z1,1, . . . , z1,n1 , . . . , zt,1, . . . , zt,nt ] be the coordinate ring of PN . Set

Q0, Q1,1, . . . , Qt,nt be the coordinate points of PN and consider the linear sub-

spaces Πi ' Pni−1 ⊂ PN defined by Πi = 〈Qi,1, . . . , Qi,ni〉, with defining ideal

IΠi = 〈z0, z1,1, . . . , z1,n1 , . . . , ˆzi,1, . . . , ˆzi,ni , . . . , zt,1, . . . , zt,nt〉.

Now let Wi be the subscheme of PN defined by the ideal Ia−aiΠi. We denote

Wi by (a− ai)Πi.

Theorem 3.6. Let Z be as above. Let Z ′ be the image of Z via the map f

and W = Z ′ +W1 + . . .+Wt ⊂ PN . Then

dim[IW ]a = dim[IZ ](a1,...,at)

where a = a1 + . . .+ at.

We want to stress again the fact that this theorem can be extremely useful

in our computation because it allows us to convert the computation of Hilbert

function of 2-fat points in multiprojective space to the computation of Hilbert

function of maybe non 0-dimensional scheme but in a standard projective space

and then in standard gradation.

Example 3.3 (Higher secant varieties of P1 × . . . × P1). We already

have considered the first trivial case P1 × P1 which is never defective.

Let’s to consider the Segre product Seg(P1 × P1 × P1) ⊂ P7. The expected

dimension for the 2-secant variety is 2 · 3 + 1 = 7, so that we expect that it fills

the ambient space. By Terracini’s lemma, we know that

dim σ2(Seg(P1 × P1 × P1)) = 7− dimkIX(1, 1, 1),


where X = {2P1, 2P2} is the scheme of two fat points in the multiprojective

space P1 × P1 × P1. Now, we can consider the scheme W in P3 given by the

f(2P1), f(2P2) and other three double points 2Q1, 2Q2, 2Q3, i.e. the linear

spaces Πi defined above. By Theorem 3.6,

dim σ2(Seg(P1 × P1 × P1)) = 7− dimkIW (3).

By Example 2.4, we know that there is no cubics with 5 singularities and

then

dim σ2(Seg(P1 × P1 × P1)) = 7,

as we expected. Thus, also P1 × P1 × P1 is non-defective.

Let’s to consider the Segre product Seg(P1 × P1 × P1 × P1) ⊂ P15. By

Proposition 3.5, we can compute the dimension of the second secant variety,

i.e.

dim σ2(Seg(P1 × P1 × P1 × P1)) = 2 · (1 + 1 + 1 + 1 + 1)− 1 = 9.

Hence, it is non-defective. Moreover, we can also find a rook covering for the

hypercube 2× 2× 2× 2 using 4 rooks as the dots in the following figure.

Figure 3.1: Rook covering of the hypercube 2× 2× 2× 2.

So that, by Proposition 3.3, we get that σ4(Seg(P1×P1×P1×P1)) fills the

ambient space and again, it is non-defective.


Now, to conclude the study of the defectiveness of the Segre product of

four copies of P1, we just have to consider the 3-secant variety. The expected

dimension of such variety is

expdim σ3(Seg(P1 × P1 × P1 × P1) = min{3 · 4 + 2, 15} = 14.

By Terracini’s lemma,

dim σ3(Seg(P1 × P1 × P1 × P1)) = 15− dimkIX(1, 1, 1, 1),

where X is a 0-dimensional scheme union of 3 general double points. By The-

orem 3.6, we can consider the scheme W in P4 given by the image of X via f

and other 4 double points and we get

dim σ3(Seg(P1 × P1 × P1 × P1)) = 15− dimkIW (4),

where W is union of 3 double points and 4 triple points. By Castelnuovo’s

Theorem 2.16, we know that 7 generic points in P4 are always on a Rational

Normal Curve.

At this point, to conclude we need to recall that the Hilbert function of

scheme of fat points with support on a Rational Normal Curve is been com-

pletely solved in a paper written by Catalisano, Elia and Gimiliano, [CEG99].

Using results of such paper, we get that the dimension of the ideal of W in

degree 4 is equal to 2, so that

dim σ3(Seg(P1 × P1 × P1 × P1)) = 13,

and hence the 3-secant variety of Seg(P1 × P1 × P1 × P1) is defective.

Moreover, Catalisano, Geramita and Gimigliano proved that actually this

is the unique defective case for Segre product of t copies of P1. Their idea

is been to use the multiprojective-affine-projective method in order to reduce

the problem to the computation of the Hilbert function of fat points in the

projective space Pt. After that, they used the idea of Horace differential method

to get the conclusion, but “how” to specialize the scheme is definitely non

trivial.


Theorem 3.7. (In [CGG05b], Theorem 2.3) The Segre product Seg(P1 ×

. . . × P1) is never defective except for the 3-secant variety of the product of

four copies of P1.

3.5 Higher secant varieties of Segre varieties

The connection between secant varieties of classical families of projective va-

rieties and applications to the study of tensor decomposition has increased

the interest to this topics. The case of symmetric tensors, i.e. homogeneous

polynomials, thanks to the Alexander-Hirschowitz Theorem, about the defec-

tiveness of Veronese varieties, is been solved, at least if we talk about the

generic symmetric tensor.

Instead, the case of tensors, which is related to the defectiveness of Segre

product, is largely unknown. Several papers in the past few years talk about

the dimension of secant varieties of Segre varieties, but, up to now, we have

only some result in particular numerical conditions.

In this short section we just want to summarize the more recent and cele-

brated results in order to stress the fact that there is a lot of work to do.

The first case is the Segre product of two projective spaces. As we have

seen in Proposition 3.5, it is completely solved.

Up to now the most complete results are due to a paper of Catalisano,

Geramita and Gimigliano [CGG05b], where the basic case of product of any

copies of P1 is completely solved, see Theorem 3.7, to a paper of Abo, Ottaviani

and Peterson [AOP09] where they characterize the defective s-varieties of the

generic Seg(Pn1× . . .×Pnt) up to s ≤ 6, see Theorem 3.8 below, and to a paper

of Catalisano, Geramita and Gimigliano of 2008 [CGG08] where they compute

the dimension of some secant varieties in the so-called unbalance case, i.e. when

the last factor of the Segre product is much bigger than the other, see Theorem

3.9 below.

3.6 Grassmann secant varieties 3. Segre varieties

Theorem 3.8. [AOP09] Consider the s-secant variety of the Segre product of

Seg(Pn1 × . . .× Pnt). The following are the only defective cases for s ≤ 6.

σ2(Seg(Pn1 × Pn2)) n1, n2 > 1

σ3(Seg(P1 × P1 × Pn)) n ≥ 3

σ3(Seg(P1 × P1 × P1 × P1))

σ4(Seg(P1 × P2 × Pn)) n ≥ 4

σ4(Seg(P2 × P2 × P2))

σ5(Seg(P1 × P2 × Pn)) n ≥ 5

σ5(Seg(P1 × P3 × Pn)) n ≥ 5

σ5(Seg(P1 × P1 × P2 × P2))

σ6(Seg(P1 × P3 × Pn)) n ≥ 6

σ6(Seg(P1 × P4 × Pn)) n ≥ 6

σ6(Seg(P2 × P2 × Pn)) n ≥ 6

σ6(Seg(P1 × P1 × P1 × Pn)) n ≥ 6

Theorem 3.9. [CGG08] Consider the Segre product Seg(Pn1 × . . .× Pnt).

Assume that nt ≥∏t−1

i=1(ni + 1)−∑t−1

i=1 ni. Then,

(i) if s ≤∏t−1

i=1(ni + 1) −∑t−1

i=1 +1, then σs(Seg(Pn1 × . . . × Pnt)) has the

expected dimension;

(ii) if∏t−1

i=1(ni + 1) −∑t−1

i=1 ni < s ≤ min{nt,∏t−1

i=1(ni + 1) − 1}, then

σs(Seg(Pn1 × . . . × Pnt)) is defective with defect s2 − s(∏t−1

i=1(ni + 1) −∑t−1i=1 ni);

(iii) if s ≥ min{nt,∏t−1

i=1(ni + 1)− 1}, then σs(Seg(Pn1 × . . .× Pnt)) = PN .

3.6 Grassmann secant varieties

In this last section we will introduce the modern definitions of Grassmann

secant varieties and Grassmann defectiveness. These notions will be similar


to the classical definitions of secant varieties that we have introduced in the

previous chapter. Grassmann secant varieties were introduce to extend a result

proved by Terracini in 1915. He showed that the defectiveness of the sth-

secant variety of the Segre product between a projective space Pk and a generic

Veronese surface V2,d is related to the defectiveness of the set of all Pk which

lies on the span of s independent points of V2.d.

Using the modern notion of Grassmann secant varieties, in 2001, Dionisi

and Fontanari proved a generalization of Terracini’s result substituting the

Veronese surfaces by a generic irreducible non-degenerate variety. In 2011,

proved the same fact using a smart rational map which explicit better the con-

nection between the Grassmann defectiveness of a projective variety V and the

defectiveness of certain Segre product Seg(Pn×V ). This result allowed to find

new interesting results about defectiveness of some Segre-Veronese varieties.

Let us start with basic definitions.

Definition 3.5. Let 0 ≤ t ≤ s − 1 ≤ n be integers and let V be a projective

variety in Pn. We define the (t,s)-Grassmann secant variety of V, say GSV (t, s),

as the closure in the Grassmannian G(t, n) of the set of all t-planes which lie

on the span of some s points of V , i.e.

GSV (t, s) :={

Λ ∈ G(t, n)∣∣ Λ⊆〈P1,...,Ps〉, whereP1,...,Ps are independent points of V

}.

Similarly to the classical definition, we have that the expected dimension is

expdim(GSV (t, s)) = min{s(dim(V )) + (t+ 1)(s− 1− t); (t+ 1)(n− t)},

hence we say that the variety V is (t, s)−defective if it has not the expected

dimension and, in such case, we define the (t, s)−defect of V as the positive

integer

δV (t, s) = expdim(GSV (t, s))− dim(GSV (t, s)).


3.6.1 The map Φ

Let V be an irreducible non-degenerate projective variety of dimension d con-

tained in Pn. For any integer h ≥ 0, we can consider the Segre embedding

ϕ : Ph × V −→ PN

([x0 : . . . : xh], [y0 : . . . : yn]) 7−→ [x0y0 : x0y1 : . . . : xhyn],

where N = (h+ 1)(n+ 1)− 1, and the Segre variety Seg(Ph× V ) ⊂ PN which

is the image of ϕ.

The map Φ = Φ(V, h, s) that we are going to construct, will be a rational

map from the secant variety σs(Seg(Ph×V )) of the Segre variety Seg(Ph×V )

into the the Grassmann secant variety GSV (h, s).

Let Λ = [λ0 : . . . : λh] ∈ Ph and P = [x0 : . . . : xn] ∈ V . For the sake of

simplicity, we use the following notation for the Segre embedding:

φ(Λ, P ) = [λ0P : . . . : λhP ].

Let A be a general point in σs(Ph × V ). Then, by definition, there exist s

distinct points Λ1, . . . ,Λs ∈ Ph and s distinct points P1, . . . , Ps ∈ V , such that

A = [a0 : . . . : aN ] ∈ 〈φ(Λ1, P1), . . . , φ(Λs, Ps)〉 .

By a suitable choice of the homogeneous coordinates Λi =[λ

(i)0 : . . . : λ

(i)h

]for all i = 1, . . . , s, we can write:

A = [a0; . . . : aN ] = ϕ(Λ1, P1) + . . .+ ϕ(Λs, Ps) =

=[λ

(1)0 P1 + . . .+ λ

(s)0 Ps : . . . : λ

(1)h P1 + . . .+ λ

(s)h Ps

],

Hence, for any general point A as above, we set:

Φ(A) :=⟨λ

(1)0 P1 + . . .+ λ

(s)0 Ps, . . . , λ

(1)h P1 + . . .+ λ

(s)h Ps

⟩.

Since A is general, the right side of this equality represents a linear space of

dimension w = min{s− 1, h}.


In this way, we get a rational map

Φ : σs(Seg(Ph × V )) 99K GSV (w, s)

A 7→ Φ(A).

We need to check that Φ is well defined. If we consider a different set

of coordinates A = [αa0 : . . . : αaN ] we have that λ(1)i P1 + . . . + λ

(s)i Ps and

α(λ(1)i P1 + . . .+λ

(s)i Ps) represents the same points for i = 1, . . . , h, so that their

spans are also equals.

Instead, suppose there exist points Mi = [µ(i)0 : . . . : µ

(i)h ] ∈ Ph and Qi for

all i = 1, . . . , s such that

A = [a0 : . . . : aN ] = φ(M1, Q1) + . . .+ φ(Ms, Qs),

and we get the equality[λ

(1)0 P1 + . . .+ λ

(s)0 Ps, . . . , λ

(1)h P1 + . . .+ λ

(s)h Ps

]=[

µ(1)0 Q1 + . . .+ µ

(s)0 Qs, . . . , µ

(1)h Q1 + . . .+ µ

(s)h Qs

].

Hence

λ(1)i P1 + . . .+ λ

(s)i Ps = α(µ

(1)i Q1 + . . .+ µ

(s)i Qs), for all i = 0, . . . , h,

and we are done.

In order to better understand how such map works and use it, now we give

a characterization of points belonging to the inverse image Φ−1(Π) of a space

Π ∈ GSV (w, s).

Lemma 3.10. Let w = min{s − 1, h} ≤ n and take a general point Π ∈

GSV (w, s). We may assume Π ⊂ 〈P1, . . . , Ps〉 for P1, . . . , Ps ∈ X distinct points

and consider B a general element of the fiber Φ−1(Π). Hence there exist points

N1, . . . ,Ns ∈ Ph such that

B = φ(N1, P1) + . . .+ φ(Ns, Ps).


Proof. by definition, we know that there are points Q1, . . . , Qs ∈ X and

M1, . . . ,Ms ∈ Ph such that

B = φ(M1, Q1) + . . .+ φ(Ms, Qs) =[µ

(1)0 Q1 + . . .+ µ

(s)0 Qs, . . . , µ

(1)h Q1 + . . .+ µ

(s)h Qs

],

and then

Φ(B) =⟨µ

(1)0 Q1 + . . .+ µ

(s)0 Qs, . . . , µ

(1)h Q1 + . . .+ µ

(s)h Qs

⟩.

Since

Π =⟨λ

(1)0 P1 + . . .+ λ

(s)0 Ps, . . . , λ

(1)h P1 + . . .+ λ

(s)h Ps

⟩and B is in the fiber over Π, it follows that each point µ

(1)i Q1 + . . . + µ

(s)i Qs

lies on Π. By the definition of w and the generality of Π, we may assume that

the points λ(1)i P1 + . . . + λ

(s)i Ps are independent. Thus, both for w = s− 1 or

for w = h, there are coefficients αi,j ∈ k, such that

µ(1)i Q1 + . . .+ µ

(s)i Qs =

w∑j=0

αi,j

(λ

(1)j P1 + . . .+ λ

(s)j Ps

)=

=

(w∑j=0

αi,jλ(j)1

)P1 + . . .+

(w∑j=0

αi,jλ(j)s

)Ps.

So, by setting ν(t)i =

∑wj=0 αi,jλ

(t)j and considering the points

Nt = [ν(t)0 : . . . : ν

(t)h ] ∈ Ph, for all t = 1, . . . , s,

we get

B =[µ

(1)0 Q1 + . . .+ µ

(s)0 Qs : . . . : µ

(1)h Q1 + . . .+ µ

(s)h Qs

]=

=[ν

(1)0 P1 + . . .+ ν

(s)0 Ps : . . . : ν

(1)h P1 + . . .+ ν

(s)h Ps

]=

= φ(N1, P1) + . . .+ φ(Ns, Ps).


3.6.2 Dimension of some secant varieties

The construction of the map Φ allows us to prove the generalization of the

Terracini’s result that we have stated previously. This section start with a

theorem which explicits the relation between the dimension of σs(Seg(Ph×V ))

and GSV (w, s).

Theorem 3.11. Let w = min{s− 1, h} ≤ n. Then we have:

dim σs(Seg(Ph × V )) = dim GSV (w, s) + (w + 1)(h+ 1)− 1.

Proof. Since Φ is a rational map, by basic theorem in algebraic geometry, to

get our statement we only need to compute the dimension of the generic fiber

of the map.

Let Π be a general element of GSV (w, s), that is a w-space contained in a

linear space 〈P1, . . . , Ps〉 with Pi independent points of V . Thus, we need to

prove that dim Φ−1(Π) = (w + 1)(h+ 1)− 1.

Even if w < h, we can obviously fix scalars λ(i)j ∈ k, with i = 1, . . . , s and

j = 0, . . . , h, such that

Π =⟨λ

(1)0 P1 + . . .+ λ

(s)0 Ps, . . . , λ

(1)h P1 + . . .+ λ

(s)h Ps

⟩.

By the definition of Φ, if we set Λi = [λ(i)0 : . . . : λ

(i)h ] ∈ Ph for all i = 1, . . . , s,

then we get that the point

A = φ(Λ1, P1) + . . .+ φ(Λs, Ps) ∈ σs(Seg(Ph ×X))

is in the fiber Φ−1(Π) and, for a general choice of the scalars, A is general.

Since s < n+ 1, we may assume, without loss of generality, that the Pi are

the points

P1 = [1 : 0 : . . . : 0], . . . , Ps = [0 : . . . : 0 : 1 : 0 : . . . : 0].

With this choice, it is easy to see that

Φ(A) =⟨[λ

(1)0 : . . . : λ

(s)0 : 0 : . . . : 0

], . . . ,

[λ

(1)h : . . . : λ

(s)h : 0 : . . . : 0

]⟩.


Now, consider another general point B ∈ Φ−1(Π). By the Lemma 3.10, there

exist s points Mi = [µ(i)0 : . . . : µ

(i)h ] ∈ Ph such that

B = φ(M1, P1) + . . .+ φ(Ms, Ps);

hence, we also get

Φ(B) =⟨[µ

(1)0 : . . . : µ

(s)0 : 0 : . . . : 0

], . . . ,

[µ

(1)h : . . . : µ

(s)h : 0 : . . . : 0

]⟩.

Since we have Φ(A) = Φ(B), it follows that:

� in the case w = k ≤ s− 1: each point[µ

(1)i : . . . : µ

(s)i : 0 : . . . : 0

]lies on

the span of the h+ 1 points[λ

(1)j : . . . : λ

(s)j : 0 : . . . : 0

], j = 0, . . . , h;

� in the case w = s− 1 < h, each point[µ

(1)i : . . . : µ

(s)i : 0 : . . . : 0

]lies on

the span of w + 1 independent points among the h+ 1 points[λ

(1)j : . . . : λ

(s)j : 0 : . . . : 0

], j = 0, . . . , h, and we may assume that they

are they correspond to j = 0, . . . , w.

In other words, there exist (w + 1)(h+ 1) elements αi,j ∈ k such that[µ

(1)i : . . . : µ

(s)i : 0 : . . . : 0

]=

w∑j=0

αi,j

[λ

(1)j : . . . : λ

(s)j : 0 : . . . : 0

],

for all i = 0, . . . , h.

Equivalently, the following linear system

M 0 0 . . . 0 0

0 M 0 . . . 0 0...

......

. . ....

...

0 0 0 . . . M 0

0 0 0 . . . 0 M

·

α0,0

α0,h

...

αh,0

αh,h

=

µ1,0

αs,0...

α1,h

αs,h

where M =

λ

(1)0 . . . λ

(1)h

λ(2)0 . . . λ

(2)h

... . . ....

λ(s)0 . . . λ

(s)h

, has solution and, more precisely, the rank of the

coefficient matrix is (w + 1)(h+ 1).


Now, since the general point of the fiber is

B = φ(M1, P1) + . . .+ φ(Ms, Ps) =

= [µ(1)0 : µ

(2)0 : . . . : µ

(s)0 : 0 : . . . : µ

(1)h : µ

(2)h : . . . : µ

(s)h : 0 : . . . : 0],

we get that, by elementary facts of linear algebra, the dimension of the fiber is

dim Φ−1(Π) = (w + 1)(h+ 1)− 1.

As a direct consequence of this Theorem, we can prove the generalization

of the Terracini’s result:

Corollary 3.12. Let h ≤ s− 1 < n. Then

V is (h, s)−defective with defect δV (h, s) = δ

if and only if Seg(Ph × V ) is s−defective with defect δs(Seg(Ph × V )) = δ.

Proof. By a direct computation, we get

expdim σs(Seg(Ph × V )) = expdim GSV (h, s) + h2 + 2h,

By the Theorem 3.11, since w = k, we also have

dim σs(Seg(Ph × V )) = dim GSV (h, s) + h2 + 2h.

Hence, since the definition of defect is the difference between the expected

dimension and the actual dimension in both cases, the s−defect of Seg(Ph×V )

and the (k, s)−defect of GSV (k, s) are the same and we are done.

Now, we can get some results about defectiveness or non-defectiveness of

sth-secant variety of Seg(Ph × V ).

Lemma 3.13. For s− 1 < h < n, we have

dim(σs(Seg(Ph × V ))) = min{s(h+ dim(V ) + 1)− 1, s(h+ n− s+ 2)− 1}.


Proof. By Theorem 3.11 we get

dim(σs(Seg(Ph × V ))) = dim(GSV (s− 1, s)) + s(h+ 1)− 1.

Now, since we have

GSV (s− 1, s) = min{s · dim(V ), s(n− s+ 1)}, 1

we are done.

Theorem 3.14. Let X ⊂ Pn be an irreducible non-degenerate projective va-

riety of dimension d.

(i) If s− 1 ≥ n, then

σs(Seg(Ph × V )) = PN ,

so it is non defective.

(ii) Let s− 1 < min{n, h},

(a) if s− 1 ≤ n− d, then

dim(σs(Seg(Ph × V ))) = s(h+ n+ 1)− 1,

and it is non defective;

(b) if s− 1 > n− d, then

dim(σs(Seg(Ph × V ))) = s(h+ n− s+ 2)− 1,

and it is defective.

(iii) If s− 1 = h < n, then

dim(σs(Seg(Ph × V ))) = min{s(h+ d+ 1)− 1, N},

and it is non defective.

1Considering s independents point on V there is only one Ps−1 lying on their span.

Consequently, the map which sends a family of s independent points to their span in the

Grassmannian of Ps−1’s contained in Pn is necessarily finite.


(iv) If h < s− 1 < n, then

dim(σs(Seg(Ph × V ))) = dim(GSX(k, s)) + h2 + 2h.

Proof. (i) Of course, it is enough to prove the case s − 1 = n. Let P1, . . . , Ps

be independent points in V , we may assume that they are coordinate points,

i.e. P1 = [1 : 0 : . . . : 0], . . . , Ps = [0 : . . . : 0 : 1].

Let A be a general point in PN where N = (h+ 1)(n+ 1)− 1, i.e.

A = [λ1,0 : . . . : λs,0 : . . . : λ1,h : . . . : λs,h].

Clearly, by definition of the Segre embedding,

A = φ(Λ1, P1) + . . .+ φ(Λs, Ps),

where Λi = (λi,0, . . . , λi,h) and we are done.

(ii-a)

N − s(h+ d+ 1)− 1 = (h+ 1)(n+ 1)− s(h+ d+ 1) ≥

(h+ 1)(s+ d)− s(h+ d+ 1) = d(h+ 1− s) > 0.

Hence, the expected dimension for σs(Seg(Ph×V )) is s(h+n+1)−1. By Lemma

3.13, since in our numerical assumption s(h+ d+ 1)− 1 ≤ s(h+n− s+ 2)− 1,

we get that the dimension is actually the expected.

(ii-b) By Lemma 3.13 and our numerical assumption, we get that

s(h+ d+ 1)− 1− dim(σs(Seg(Ph × V ))) =

s(h+ d+ 1)− 1− (s(h+ n− s+ 2)− 1) = s(d− n+ s− 1) > 0;

and also

N − dim(σs(Seg(Ph × V ))) =

(h+ 1)(n+ 1)− 1− (s(h+ n− s+ 2)− 1) = (n− s+ 1)(d− s+ 1) > 0.

Hence the expected dimension is bigger than the actual dimension and

σs(Seg(Ph × V )) is defective.

(iii) For s− 1 = h, we get

s(h+ n− s+ 2)− 1 = s(r + 1)− 1 = (h+ 1)(r + 1)− 1 = N,


hence we get the conclusion directly by Lemma 3.13.

(iv) By Theorem 3.11.

Corollary 3.15. If h = s− d, then σs(Seg(Ph × V )) is never defective.

Proof. Assume s − 1 = h. In this case by Theorem 3.14 (iii), the sth-secant

variety of Seg(Ph × V ) is never defective, i.e.

dim(σs(Seg(Ph × V ))) = min{s(d+ s)− 1, (h+ 1)(n+ 1)− 1}.

Moreover, since n− d = s− 1, we have

s(d+ s)− 1 = s(n+ 1)− 1 = (h+ 1)(n+ 1)− 1.

Hence, we get that in general for s 6= h+ 1:

� if s > h+ 1, dim(σs(Seg(Ph × V ))) = N ;

� if s < h+ 1, dim(σs(Seg(Ph × V ))) = s(h+ d+ 1)− 1;

and we are done.

Example 3.4. Using these numerical conditions which guarantee the defec-

tiveness or non-defectiveness of higher secant varieties of the Segre product

Seg(Ph×X), we can easily understand the non-defectiveness of some interest-

ing Segre-Veronese varieties.

(i) Consider Y the Segre-Veronese embedding of P(n+12 ) × Pn of bi-degree

(1, 2). Then σsY is never defective.

By definition, if X is the image of the Veronese embedding of Pn of degree

2 in P(n+22 )−1, we have

Y = Seg(P(n+2

2 ) ×X).

Since(n+1

2

)=(n+2

2

)− 1− n, by Corollary 3.15, we are done.

(ii) Similarly, let Y be the Segre-Veronese embedding of Ph×Pn1×. . .×Pnt of

multi-degree (1, d1, . . . , dt). If h =∏t

i−1

(ni+dini

)−∑t

i=1 ni−1, by Corollary

3.15 we get again that σs(Y ) is never defective. [EH00]

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