bounds on hilbert functions - simple...

Bounds on Hilbert Functions

ORNELLA GRECO

Licentiate ThesisStockholm, Sweden 2013

TRITA-MAT-13-MA-03ISSN 1401-2278ISBN 978-91-7501-901-7

KTHInstitutionen för Matematik

100 44 StockholmSWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolanframlägges till offentlig granskning för avläggande av teknologie licentia-texamen i matematik måndagen den 11 november 2013 kl 10.00 i sal 3418,Kungl Tekniska högskolan, Lindstedtsvägen 25, Stockholm.

c© Ornella Greco, 2013

Tryck: Universitetsservice US AB

iii

Abstract

This thesis is constituted of two articles, both related to Hilbertfunctions and h-vectors. In the first paper, we deal with h-vectorsof reduced zero-dimensional schemes in the projective plane, and, inparticular, with the problem of finding the possible h-vectors for theunion of two sets of points of given h-vectors. In the second paper, wegeneralize the Green’s Hyperplane Restriction Theorem to the case ofmodules over the polynomial ring.

iv

Sammanfattning

Denna avhandling består av två artiklar, båda relaterade till Hil-bert funktioner och h-vektorer. I den första artikel hanterar vi h-vektorer för reducerade noll-dimensionella scheman i det projektivaplanet, och, i synnerhet, med problemet att finna möjliga h-vektorerför en union av två mängder av punkter med givna h-vektorer. I denandra artikeln, generaliserar vi Greens Hyperplansrestriktionssats tillfallet av moduler över polynomringar.

Contents

Contents v

Part I: Introduction and summary

1 Introduction 11.1 Hilbert functions and h-vectors . . . . . . . . . . . . . . . . . 21.2 Lexicographic Ideals and Modules . . . . . . . . . . . . . . . . 31.3 Bounds on the Hilbert function . . . . . . . . . . . . . . . . . 81.4 The h-vector of a reduced zero-dimensional scheme in P2 . . . 12

2 Summary of results 172.1 Paper A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Paper B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

References 21

Part II: Scientific papers

Paper AThe h-vector of the union of two sets of points in the projective

plane.

Paper BGreen’s Hyperplane Restriction Theorem: an extension to modules.

v

Acknowledgements

First of all, I would like to thank my supervisor, Mats Boij, for the support,for his scientific advises and also for giving me the chance to visit him atMSRI.I also would like to express my gratitude to Ralf Fröberg, for being presenteven before the start of my PhD studies, and for giving me always veryappreciated suggestions and comments.A deep Sepas also to my Iranian support, Sadna and Afshin! And big Grazieto my Italian cheer, Alessandro and Martina!I would like to thank also all the people met at MSRI, in particular Emanuele,Antonello, Emma, Ali, Jack and Jonathan, they all contributed to make melive an amazing period in Berkeley.Last but not least, I must thank Ivan, for being everything to me, and fi-nally my parents, Saro and Annamaria, for being such good listeners andsupporters.

vii

Chapter 1

Introduction

This thesis is constituted of two papers. The topic of both papers has beenthe study of an important invariant in commutative algebra: the Hilbertfunction, and, in relation with it, the h-vectors of rings and modules.

In the first paper (see [G-M-S2]), we study the Hilbert functions of setsof points in P2: namely we consider the coordinate ring associated with aset of points, and then calculate the first difference of the Hilbert functionof this ring. In this way, we get a vector of finite length, called the h-vectorof the set of points, which encodes information about the configuration ofthe set of points.Then, we partially answer to the following question: given the h-vectors oftwo sets of points in the projective plane (there can be many configurationsof points with the same h-vector), what are all possible h-vector for theunion of two sets of points associated with the given h-vectors? We give twokinds of bounds on the h-vectors, and we also provide an algorithm thatcalculates many possible h-vectors, but we are not able to determine if theyare all.

In the second paper, we study Hilbert functions in a more general set-ting. Namely, we generalize to the case of modules the Hyperplane Re-striction Theorem proved by M. Green in [Gr], which gives a bound on theHilbert function of a general linear section of a symmetric algebra in termsof Macaulay’s binomial representations and lexicographic ideals.

1

2 CHAPTER 1. INTRODUCTION

In the next sections of this chapter, we are going to summarize thedifferent concepts that are needed for the understanding of the two papers.

1.1 Hilbert functions and h-vectors

Now we are going to give an overview on the basic concepts of Hilbertfunctions, Hilbert series, and h-vectors. First of all, let us provide the settingin which we are going to work.

Let us consider a field k. A standard graded k-algebra S = ⊕i≥0Si is afinitely generated k-algebra with all generators in degree 1. The main exam-ple of such an algebra is the polynomial ring, k[x1, . . . , xn], with the standardgrading, or a quotient of the polynomial ring, k[x1, . . . , xn]/I, where I is astandard graded ideal of k[x1, . . . , xn].

Let M be a graded finitely generated S-module, and denote by Mi thei-th graded component.

Definition 1.1.1. The Hilbert function ofM is a numerical function definedby H(M,d) = dimk(Md). The Hilbert series of M is defined as HM (t) =∑i∈ZH(M, i)ti.

Here we give a survey on some properties of Hilbert functions and Hilbertseries. There are more complete surveys on this topic: see, for example,Chapter 4 in [B-H], or Chapter 6 in [H-H].

Theorem 1.1.2. [H-H, Theorem 6.1.3] Let d be the Krull dimension of M ,then:

• there exists a Laurent polynomial QM ∈ Z[t, t−1] such that HM (t) =QM

(1−t)d ;

• there exists a polynomial PM ∈ Q[x] of degree d−1 such that H(M, i) =PM (i) for all i > deg(QM )− d (i.e. the Hilbert function is of polyno-mial type).

From the previous theorem, we derive the following definitions.

Definition 1.1.3. The polynomial PM is called the Hilbert polynomial ofM .Let us write QM =

∑bi=a hit

i for a, b ∈ Z, then the vector (ha, ha+1, . . . , hb)is the h-vector of M .

1.2. LEXICOGRAPHIC IDEALS AND MODULES 3

The theorem below relates Hilbert functions and graded Betti numbers.

Theorem 1.1.4. [B-H, Lemma 4.1.13] Let M be a finitely generated S-module of finite projective dimension, and let us give a graded free resolutionof M :

0 −→ ⊕jS(−j)βpj −→ · · · −→ ⊕jS(−j)β0j −→M −→ 0,

then HM (t) = SM (t)HS(t), where SM (t) =∑i,j(−1)iβijtj, and, in particu-

lar, if S = k[x1, . . . , xn], HM (t) = SM (t)(1−t)n .

1.2 Lexicographic Ideals and ModulesIn this section, we will introduce the concepts of monomial orders, initialideals and lexicographic ideals. Then we will extend these concepts to freeS-modules. Let S denote the polynomial ring k[x1, . . . , xn].

Definition 1.2.1. A monomial order on S (or equivalently on Nn) is a totalorder, <, on the set of monomials in S, Mon(S), such that:

• if u, v ∈ Mon(S) and u < v, then uw < vw for all w ∈ Mon(S).

• < is a well ordering on Nn.

Well known examples of monomial orders, where we have x1 > x2 > · · · >xn, are:

• the pure lexicographic order (lex), in which xa <lex xb iffthe first left non zero component in a − b is negative;

• the graded lexicographic order (deglex), defined by

xa <deglex xb ⇔ |a| < |b| or |a| = |b| and a <lex b.

• the reverse lexicographic order (revlex), where it is defined xa <revlexxb iff either |a| < |b| or |a| = |b| and last right non zero componentin a − b is positive.

Example 1.2.2. In N3 we have that a = (1, 0, 0) >lex (0, 3, 2) = b, since(0, 3, 2)− (1, 0, 0) = (−1, 3, 2), but since |a| < |b|, we have that a <deglex b.Moreover, c = (1, 2, 2) > b in all three monomial orders, since they havesame total degree, and c − b = (1,−1, 0), since the first non zero entry ispositive, and the last non zero entry is negative.


Let us fix a monomial order in S, <, and let u be a monomial in degree d.Then, we give the following notations:Lu = {v ∈ Mon(Sd)| v < u}, Ru = {v ∈ Mon(Sd)| v ≥ u}.

Definition 1.2.3. A monomial ideal I in S is called lexicographic ideal if,for every degree d, Id is a lex-segment, i.e. a saturated chain (with respectto the chosen monomial order) of monomials starting from the largest, i.e.if there is a monomial u ∈ Sd such that Id = Ru.

Given a monomial order <, we can order the terms of a polynomial f ∈ Sin a unique way, and in particular we can consider the leading (or initial)term of f , denoted by in<(f).

Definition 1.2.4. Given an ideal I in S the initial ideal of I with respect to< is the ideal generated by the initial terms (with respect to the monomialorder <) of the polynomials in I. We denote this ideal by in<(I).

The initial ideals have nice properties, in particular we have many resultsthat compare the invariants of graded ideals and their initial ideals.

Proposition 1.2.5. [H-H] Let I be a graded ideal of S, and let < be amonomial order on S. Then S/I and S/in<(I) have the same Hilbert func-tion.

Theorem 1.2.6. [H-H, Theorem 3.3.4] Let I be a graded ideal of S and <a monomial order on S. Then the following properties hold:

1. dim(S/I) = dim(S/in<(I));

2. projdim(S/I) ≤ projdim(S/in<(I));

3. reg(S/I) ≤ reg(S/in<(I));

4. depth(S/I) ≥ depth(S/in<(I));

Generic Initial ideals

We now define the concept of generic initial ideal. Let k be an infinite field,and again let us denote by S the polynomial ring, k[x1, . . . , xn].

Consider the general linear group, GL(n), i.e. the group of all invertiblen×nmatrices with entries in k. This group can be equipped with the Zariski


topology, inherited from kn×n; moreover, GL(n) is Zariski open: this impliesthat a subset of GL(n) is open if and only if it is a Zariski open set in kn×n.Let us consider the action of GL(n) on the polynomial ring S: a matrix(aij) ∈ GL(n) sends xj to

∑ni=1 aijxi; this action can be then extended to

any f ∈ S.We have the following result.

Theorem 1.2.7. [H-H, Theorem 4.1.2] Let I be a graded ideal in S, and< a monomial order on S, then there exists a non-empty Zariski open setU ⊆ GL(n), such that ∀ g, h ∈ U , in<(gI) = in<(hI).

Definition 1.2.8. The ideal in<(gI), where g ∈ U , is called generic initialideal of I, with respect to <, and denoted by gin<(I) (or simply gin(I),when the monomial order is clear).

We now discuss some properties of generic initial ideals, namely they areBorel-fixed, and, in certain characteristic of the field k, they are stronglystable.Let us consider the Borel subgroup of GL(n), denoted by B(n), i.e. thegroup of all upper triangular invertible n× n matrices. From the definitionof the action, a matrix in B(n) will send x1 to a multiple of itself, and xn tothe linear combination of all variables xi’s with coefficients the last columnof tha matrix.

Definition 1.2.9. An ideal I in S is Borel-fixed if gI = I for every matrixg ∈ B(n).

Definition 1.2.10. A monomial ideal I in S is strongly stable if for everymonomial u ∈ I and for j such that xj divides u, we have that xi(u/xj) ∈ I,∀ i < j.

Theorem 1.2.11. [H-H, Theorem 4.2.1] Let I be a graded ideal in S and< a monomial order on S, then gin<(I) is a Borel-fixed ideal.

The following theorem relates Borel-fixed ideals with strongly stable ideals.

Theorem 1.2.12. [H-H, Proposition 4.2.4]

• Let I be a graded ideal in S, if I is Borel-fixed, then it is a monomialideal;

• If I is strongly stable, then it is Borel-fixed;


• Let I be a Borel-fixed ideal, and let a be the biggest exponent appearingamong the generators of I, then, if char(k) = 0 or char(k) > a, I isstrongly stable.

Definition 1.2.13. We say that a property P holds for a generic linearform ` if there is a non-empty Zariski open set U ⊆ S1 such that P holdsfor all ` ∈ U .

In case we choose the monomial order < to be the reverse lexicographicorder, we have the following result:

Proposition 1.2.14. [Gr2, Corollary 2.15] Let h be a generic linear formand I a graded ideal in F , then gin<(Ih) = gin<(I)xn.

Lexicographic Modules and Generic Initial Modules

Let us extend these definitions to free S-modules.Let F be a free finitely generated S-module and let us fix an homogeneousbasis {e1, e2, . . . , er} and let deg(ei) = fi, where, without loss of generality,we may assume that f1 ≤ f2 ≤ · · · ≤ fr.We now define the monomial modules and we induce a lexicographic orderon F in such a way that the concept of lexicographic module may be defined.

Definition 1.2.15. A monomial in F is an element of the form mei wherem ∈ Mon(S).A submodule M ⊆ F is monomial if it is generated by monomials, in thiscase it can be written as I1e1 ⊕ I2e2 ⊕ · · · ⊕ Irer, where Ii is a monomialideal for i = 1, 2, . . . , r.

Let us now extend the concept of monomial order to modules, in particularlet us define the graded lexicographic order and the reverse lexicographicorder on F .

Definition 1.2.16. Given two monomials in F , mei and nej , we say thatmei >deglex nej if either i = j and m >deglex n in R or i < j. In particular,we have that e1 > e2 > · · · > er.

Definition 1.2.17. A monomial submodule L is a lexicographic moduleif, for every degree d, Ld is spanned by the largest, with respect to thelexicographic order, H(L, d) monomials.


Definition 1.2.18. The reverse lexicographic order on F is defined bychoosing an order on the basis of F , say e1 > · · · > er and by settingmei >revlex nej iff either deg(mei) > deg(nej) or the degrees are the sameand m >revlex n in S or m = n and i < j.

Remark 1.2.19. The graded lexicographic order is a "position over term"monomial order, on the contrary the reverse lexicographic order is "termover position" monomial order.

Definition 1.2.20. The initial module of a submodule M with respect toa chosen monomial order on F , denoted by in<(M), is the submodule ofF generated by the set {in<(m)| m ∈ M}, i.e. by all the leading terms ofelements in M .

In case we choose the reverse lexicographic order, we have some nice results.Namely:

Proposition 1.2.21. [Ei, Proposition 15.12] Suppose that F is a free S-module with basis {e1, . . . , er} and reverse lexicographic order. Let M be agraded submodule. Then:

• in(M + xnF ) = in(M) + xnF ;

• (in(M) :F xn) = in(M :F xn).

Let us recall the concept of generic initial module. In his thesis, see[Pa], K. Pardue defined this concept in a more general setting, here we willrestrict to the case is needed in the second paper.Let us consider the group GL(F ) of the S-module automorphisms of F . Anelement φ in GL(F ) is a homogeneous automorphism and can be representedby a matrix (tij), with tij ∈ Sfi−fj

, where φ(ei) =∑rj=1 tijej . Moreover,

the general linear group GL(n) acts on F k-linearly; and we have also anaction of GL(n) on GL(F ), given by a · φ = aφa−1, for all a ∈ GL(n) andφ ∈ GL(F ).So, let us denote by G the semidirect product G = GL(n) o GL(F ).As done for the ideals, let us consider the Borel subgroup of G, B, which isobtained throught the semidirect product of B(n), the subgroup of GL(n)of upper triangular invertible matrices, and B(F ), the subgroup of all au-tomorphisms in GL(F ) represented by lower triangular matrices (they sendeach el to an S-linear combination of e1, . . . , el).


In Pardue’s thesis, there is the generalization to modules of many results ongeneric initial ideals. In particular, we have the following result.

Proposition 1.2.22. [Pa, Chapter 1] Let M ⊂ F be a graded module, andlet < be a monomial order on F , then there exists a Zariski open set U ⊆ Gsuch that in<(φM) is constant for every φ ∈ U . Moreover, in<(φM) is fixedby the action of group B.

Definition 1.2.23. The monomial submodule in<(φM), φ ∈ U , is calledthe generic initial module of M , and denoted by gin<(M).

Theorem 1.2.24. [Pa, Proposition 5, Chapter 1] A submodule M ⊆ F isfixed by B if and only if:

• M is a monomial submodule, i.e. can be written as I1e1 ⊕ · · · ⊕ Irer;

• the ideals Ij are Borel-fixed;

• for every i < j, (x1, . . . , xn)fj−fiIj ⊆ Ii.

1.3 Bounds on the Hilbert function

Let us now recall some results on extremal properties of Hilbert functions,and bounds on Hilbert functions. One of the most important result in thisarea is Macaulay’s theorem (see [Ma, B-H, K-R]), which characterizes thepossible Hilbert functions of homogeneous k-algebras. Before announcingthis theorem, we need the concept of Macaulay’s representations of integers.

Definition 1.3.1. Given a, d ∈ N, the d-th Macaulay representation of a isthe only way of writing a in the following way:(

add

)+(ad−1d− 1

)+ · · ·+

(a11

),

where ad > ad−1 > · · · > a1 ≥ 0.

Observation 1.3.2. If a =(add

)+(ad−1d−1

)+ · · · +

(a11)and b =

(bdd

)+(bd−1d−1

)+

· · ·+(b1

1), a ≥ b if and only if (ad, . . . , a1) ≥lex (bd, . . . , b1).

If c < d, then(cd

)= 0.

1.3. BOUNDS ON THE HILBERT FUNCTION 9

Moreover, we introduce three numerical functions, defined by means of thed-th Macaulay representation of a =

(add

)+(ad−1d−1

)+ · · ·+

(a11), namely:

a〈d〉 =(ad + 1d+ 1

)+(ad−1 + 1

d

)+ · · ·+

(a1 + 1

2

),

a〈d〉 =(ad − 1d

)+(ad−1 − 1d− 1

)+ · · ·+

(a1 − 1

1

),

a(d) =(

add+ 1

)+(ad−1d

)+ · · ·+

(a12

).

The growth of the lexicographic ideals is related to the first function,namely:

Proposition 1.3.3 (Macaulay). [B-H, Proposition 4.2.9] Let I be a homo-geneous ideal in S, and d ∈ N.Then

H(S/I, d+ 1) ≤ H(S/I, d)〈d〉,

and equality holds if Id is a lex-segment space and Id+1 = S1 · Id.

The following result gives us the characterization of the Hilbert function.

Theorem 1.3.4 (Macaulay). [B-H, Theorem 4.2.10] Let k be a field, andh : N→ N. The following conditions are equivalent:

1. there is a homogeneous k-algebra S with Hilbert function H(S, d) =h(d);

2. there is a homogeneous k-algebra with monomial relations S with Hilbertfunction H(S, d) = h(d);

3. h(0) = 1, h(d+ 1) ≤ h(d)〈d〉 for all d ≥ 1.

Another classical result in this topic is the Hyperplane Restriction The-orem, proved by Green in [Gr], which gives a bound for the codimension ofthe generic linear restriction of a vector space generated in a certain degreeby the codimension of a lex-segment space with same degree and dimension.This result was also used by Green to give a new, less technical, proof ofMacaulay’s Theorem.


Let ` ∈ S1 be a generic linear form, and I a homogeneous ideal, let usdenote by (S/I)` the reduction to this hyperplane, i.e. the ring S/[I + (`)].Similarly, if V ⊆ Sd a lex-segment space, we denote by V` the image of V inS/(`).

Proposition 1.3.5. [K-R, Proposition 5.5.23] Let k be an infinite field,d ∈ N, V ⊆ Sd a lex-segment space. Then:

1. For a generic linear form ` ∈ S1, there is a homogeneous linear changeof coordinates φ : S → S such that φ(V ) = V and φ(`) = xn;

2. For a generic linear form ` ∈ S1, we have

codimk(V`) = codimk(Vxn) = codimk(V )〈d〉.

Theorem 1.3.6 (Green). [B-H, Theorem 4.2.12] Let I be a homogeneousideal in S, d ∈ N, then

H((S/I)`, d) ≤ H(S/I, d)〈d〉,

where ` is a generic linear form. Equality holds if Id is a lex-segment space.

The analogous result of Macaulay’s Theorem for the exterior algebra wasproved by Kruskal and Katona (see [Ka, Kr]), and it provides a characteri-zation of f -vectors of simplicial complexes.

Let k be a field,W an n-dimensional k-vector space with basis {e1, . . . , en}.Let E = ⊕i ∧i W be the exterior algebra. A monomial in E is an elementeG = ej1 ∧ · · · ∧ eji , where G = {j1 < · · · < ji} ⊆ [n].In a natural way we can define the lexicographic order over the monomialsin the exterior algebra: eG <lex eL iff xG <lex xL in S.

Let V = {v1, . . . , vn} be a finite set. A simplicial complex ∆ on V is acollection of subsets of V such that {vi} ∈ ∆ for every i, and if G ∈ ∆ andF ⊆ G, then F ∈ ∆. The elements of ∆ are called faces of ∆. For a givenface F ∈ ∆, dim(F ) = |F | − 1.Given a simplicial complex ∆, for all i = −1, 0, . . . ,dim(∆), let fi be thenumber of i-dimensional faces of ∆. Note that ∅ ∈ ∆, so f−1 = 1 andf0 = |V |.

Here we state a theorem which characterizes the possible f -vectors forsimplicial complexes.

1.3. BOUNDS ON THE HILBERT FUNCTION 11

Theorem 1.3.7 (Kruskal-Katona). A vector (f−1, f0, . . . , fd−1) ∈ Zd+1 isthe f -vector of some (d− 1)-dimensional simplicial complex if and only if

0 < fi+1 ≤ f (i+1)i , ∀ 0 ≤ i ≤ d− 2.

Given a simplicial complex ∆ over V = {v1, . . . , vn}, we can associate amonomial ideal in the exterior algebra E, in the following way:

ei1 ∧ · · · ∧ eit ∈ I∆ ⇔ {vi1 , . . . , vit} /∈ ∆.

The algebra E/I∆ is called Stanley-Reisner algebra, and it is denoted byk{∆}. Notice that

fd(∆) = H(k{∆}, d),this lets us interpret the Kruskal-Katona Theorem as a Macaulay bound onthe growth of the Hilbert functions for exterior algebra.

Macaulay’s Theorem and Green’s Hyperplane Restriction Theorem havebeen extended to modules by many authors, in particular by Hulett in [Hu]and Gasharov in [Ga]. Here we recall their results.

Theorem 1.3.8. [Hu, Theorem 1] Let H be the Hilbert function of a gradedquotient F/M , where M is a homogeneous submodule of F . Then there isa lexicographic module L ⊆ F such that H(F/L, d) = H(d) for all d, andβi,j(F/L) ≥ βi,j(F/M).

As a consequence, we get:

Theorem 1.3.9. [Hu, Corollary 6] Let M be a graded submodule of F , then∃ N ≤ t such that we have the unique expression for

H(F/M, d) =N−1∑i=1

(n+ d− fi − 1

d− fi

)+(

a0d− fN

)+ · · ·+

(as

d− fN − s

),

where( a0d−fN

)+ · · ·+

( as

d−fN−s)<(n+d−fN−1

d−fN

), and it is the Macaulay repre-

sentation of a lex-segment in degree d− fN in the N -th component of F .Moreover,

H(F/M, d+1) ≤N−1∑i=1

(n+ d− fid− fi + 1

)+(

a0 + 1d− fN + 1

)+· · ·+

(as + 1

d− fN − s+ 1

),

where the bound is achieved by the lexicographic submodule of the previoustheorem.


In the following result, due to Gasharov, he describes a bound on thegrowth of Hilbert function of modules, and on the generic hyperplane sec-tion. Let again F = ⊕ti=1Sei, and deg(ei) = fi, with fi ≥ fi+1.

Theorem 1.3.10. [Ga, Theorem 4.2] Let N ⊆ F be a graded submodule,then ∀ p ≥ 0 and ∀ d ≥ p+ f1 + 1 we have that:

1. H(F/N, d+ 1) ≤ H(F/N, d)〈d−f1−p〉;

2. H(F`/N`, d) ≤ H(F/N, d)〈d−f1−p〉, where ` is a generic linear form.

1.4 The h-vector of a reduced zero-dimensionalscheme in P2

The study of sets of points in the projective plane constitute a really im-portant topic in algebra. There are several surveys dealing with it (see[Ei2, Gr2]). This section will give a summary of some results and definitionsuseful to understand the results contained in the first paper.

Let k be an algebraically closed field, P2 = P2(k) be the projectiveplane over k and S = k[x0, x1, x2] its homogeneous coordinate ring. In thissection, we will refer to a finite set of distinct points in the projective planeas a reduced zero-dimensional scheme.

If X is a zero-dimensional scheme in P2(k), we denote by HX(i) =dimk((S/IX)i) its Hilbert function and by h = (h0, . . . , ht) its h-vector whereh0 = 1 and hi = 4HX(i) = HX(i)−HX(i− 1) ∀ i > 0.In case X is reduced, the number of points in X, also called degree of X, isgiven by h0 + · · ·+ ht.It is important to point out that to a given h-vector may correspond sev-eral configurations of points, i.e. several different reduced zero-dimensionalschemes.

Remark 1.4.1. To a given h-vector, (h0, h1, . . . , ht), we can assign a diagramby drawing columns of hi boxes for all i. For example if h = (1, 2, 3, 2, 2, 1),we will draw the following diagram.

1.4. THE H-VECTOR OF A REDUCED ZERO-DIMENSIONALSCHEME IN P2 13

We now define three measures of the h-vectors.Let X be a reduced zero-dimensional scheme and let h be its h-vector, thenthe length of h is τ(h) + 1, where:

τ(h) = max{i | hi 6= 0} = min{i | HX(i) = HX(i+ 1)}.

We define b(h) = max{hi| i = 0, 1, . . . , τ(X)} as the height of the h-vectorh, which is also equal to the least degree of a generator of IX , the vanishingideal of X.Sometimes we will use τ(X) and b(X) instead of τ(h) and b(h).We define also η(h, d) :=

∑τ(X)i=0 min{hi, d}. If d is greater or equal to the

height of h, then η(h, d) is just equal to the degree of the scheme.

The following theorem due to E. D. Davis (see [Da]) provides a descrip-tion of the structure of the h-vector of a reduced zero-dimensional scheme,and gives also information about the configuration of the sets of points as-sociated to an h-vector.

Theorem 1.4.2 (Davis). The h-vector (h0, . . . , hτ(X)) of a reduced zero-dimensional scheme X ⊂ P2 satisfies the following conditions:

1. hd = d+ 1, for d = 0, 1, . . . , b− 1, and hb ≤ b;

2. hd+1 ≤ hd, for d ≥ b− 1;

3. If hd = hd+1 = e for some d ≥ b− 1, the generators of IX of degree atmost d+1 have a common factor of degree e. This leads to a partitionof X into X1 ∪X2, where X1 lies on a curve of degree e and X2 hasthe h-vector given by (he − e, he+1 − e, . . . , hd−1 − e) .Where b is the least degree of a generator of IX .

In the case described in part (3) of the theorem, we say that the h-vector ofX has a flat of height e.

It is possible to introduce a natural partial ordering in the set of theHilbert functions of reduced zero-dimensional scheme in the projective plane

H := {4HX | X ⊆ P2 is a finite set of points}.

Namely, let H1 = (HX(i))i∈N and H2 = (HY (i))i∈N be two Hilbert functions,we will say that H2 is more generic than H1, and we will write H1 ≤g H2,


if HX(i) ≤ HY (i), ∀ i ∈ N. We say also in this situation that H1 is morespecial than H2.This partial order induce also a partial order (with the same notation, ≤g)on the h-vectors. Indeed, if (h0, h1, . . . , hs) and (h′0, h′1, . . . , h′t) are the h-vectors of two finite sets of points in P2. Then:

(h0, h1, . . . , hs) ≤g (h′0, h′1, . . . , h′t) :⇐⇒j∑i=0

hi ≤j∑i=0

h′i, ∀ j = 0, . . . , s

Going to the diagram of an h-vector, this means that, by moving one boxfrom a row to an upper row in such a way that the result is admissible (i.e.satisfies the conditions in Theorem 1.4.2), we get a more generic h-vector.

In the first paper, we provided two algorithms that build configurationsof points associated to an h-vector, and that calculate a range of possibleh-vectors for the union. In both of them, we have used heavily the theoryof 2-type vectors, pseudo type vectors and linear configurations, that we arenow going to introduce. This theory is fully described in [G-M-S1]

Definition 1.4.3. A 2-type vector is a vector (d1, d2, . . . , dt), where 0 <d1 < d2 < · · · < dt .

To any h-vector, associated to a reduced zero-dimensional subscheme of P2,corresponds only one 2-type vectors. The following theorem explains thiscorrespondence.

Theorem 1.4.4. [G-M-S1, Theorem 2.4, Theorem 2.5] Let S2 denote thecollection of Hilbert functions of all reduced zero-dimensional schemes inP2. Then, there is a 1-1 correspondence between S2 and the set of 2-typevectors. Moreover, let T = (d1, d2, . . . , dt) be a 2-type vector , and Hi theHilbert function of di collinear points. Then, T corresponds to the Hilbertfunction defined by H(j) = Ht(j) + · · ·+H1(j − (t− 1)).

Once we have the definition of 2-type vector we can define the concept oflinear configuration.

Definition 1.4.5. Let T = (d1, . . . , dt) be a 2-type vector. Let L1, . . . , Ltbe t distinct lines in P2 and Xi a set of di distinct points on Li, for alli = 1, . . . , t. Moreover, we suppose that, for i 6= j, Li does not contain anypoint of Xj . Then, X = ∪ti=1Xj is called a linear configuration of type T .

1.4. THE H-VECTOR OF A REDUCED ZERO-DIMENSIONALSCHEME IN P2 15

The following result shows that the Hilbert function associated to a linearconfiguration of a given type T depends only on the type, and not on thechoice of the lines and of the points on them.

Theorem 1.4.6. [G-M-S1, Theorem 2.8] Let X be a linear configuration oftype T , and let H be the Hilbert function associated to T . Then the Hilbertfunction of X, HX is H.

Remark 1.4.7. Given a reduced zero-dimensional scheme which is also alinear configuration, its type is nothing else than a partition of the degreeof the scheme constituted by strictly increasing positive integers.A similar definition of linear configurations can be given for partitions ofthe degree of the scheme constituted by non-decreasing positive integers.

Definition 1.4.8. A pseudo type vector is a sequence of positive integersT = (d1, . . . , dt), where di ≤ di+1 ∀ i, and if di−1 = di, then di < di+1.

A pseudo linear configuration of type T is a set of points X =t⋃i=1

Xi, whereXi is a set of di distinct points on a line Li. The lines L1, . . . , Lt are alldifferent, and none of the points of Xi lies on Lj for i 6= j.

Also in this case, an O-sequence can be associated to a pseudo type vec-tor (see [G-M-S1]), but in general the Hilbert function of a pseudo linearconfiguration of type T = (d1, . . . , dt) is not uniquely determined, unless itsatisfies the following:

between any two zero entries of ∆T there is at least one entry > 1. (1.1)

To a given h-vector, (h0, . . . , ht), one can associate a monomial ideal Isuch that the standard graded k-algebra S/I has the given h-vector.

1. From the h-vector (h0, . . . , ht) we pass to its geometric representationby drawing hi boxes for all i.

2. By Davis’s Theorem we have that hi = i + 1 for i = 0, . . . , b − 1 andhi ≥ hi+1 ∀i ≥ b − 1. Denote by di the number of squares in thei-th row, for i = 1, . . . , b. Notice that the vector D = (d1, . . . , db) is a2-type vector.

3. Let I be the ideal generated by xdb , xdb−1y, . . . , yb. Then the k-algebraS/I has h-vector (h0, h1, . . . , ht). Note also that the ideal I is a lexi-cographic ideal.


To the ideal I = (xdb , xdb−1y, . . . , yb) we can assign a set of points in P2,whose defining ideal has the same h-vector. This set of points is a linearconfiguration of type D = (d1, . . . , db). In order to get it we first choosetwo sets of distinct elements in k, {α1, . . . , αdb

} and {β1, . . . , βb}, and thenreplace every generator xiyj , of I by (x−α1z) . . . (x−αiz)(y− β1z) . . . (y−βjz).In the special case where α1 = 0, . . . , αdb

= db−1 and β1 = 0, . . . , βb = b−1,we get the defining ideal of the following set of points:

db points with coordinates (i : 0 : 1), i = 0, . . . , db − 1;db−1 points with coordinates (i : 1 : 1), i = 0, . . . , db−1 − 1;...d1 points with coordinates (i : b− 1 : 1), i = 0, . . . , d1 − 1.

We will call this the standard linear configuration of type D.

Finally, we need the definition of sum of two partitions.Remark 1.4.9. Given a partition of a number n, i.e. (c1, . . . , ct), whereci ≤ ci+1 and c1 + · · · + ct = n, then clearly by adding zero entries we willstill get a partition of n. If the first partition is associated to a scheme, soare the ones with the zeros.

Definition 1.4.10. Let c = (c1, . . . , ct) and d = (d1, . . . , dv) be two par-titions of n respectively m. Assume in addition that at least one of thosepartitions is either a 2-type or a pseudo type vector whose first differencesatisfies the Condition (1.1). We say that a partition of n+m is the sum ofc and d, if it is obtained by ordering the sequence

{ci + dj}i=1,...,t;j=1,...,v

(where each ci and dj appear exactly once in the sums) in a non-decreasingway.

Chapter 2

Summary of results

2.1 Paper AThe h-vector of the union of two sets of points in the projective plane.

The first paper deals with the problem of finding all the possible h-vectorsof the union of two sets of points with given h-vectors h and h′.In order to exclude an h-vector from the set of the possible ones, we foundsome bounds on the main units of measure, which are b, τ and η.Namely, we proved the following inequalities:

Theorem 2.1.1. Given two sets of points X and Y in P2, with give h-vectors respectively h and h′ we have that

max{τ(X), τ(Y )} ≤ τ(X ∪ Y ) ≤ τ(X) + τ(Y ) + 1.

Theorem 2.1.2. Given two sets of points X and Y in P2, we have thefollowing bounds for the height of the resulting h-vector:

max{b(X), b(Y )} ≤ b(X ∪ Y ) ≤ min{b(X) + b(Y ), b(G)}

where G consists of deg(X ∪ Y ) generic points.

Theorem 2.1.3. Let h be the h-vector of X with flat of height r, h′ theh-vector of Y with flat of height s and h′′ the h-vector of X ∪ Y . Then wehave

η(h′′, r + s) ≥ η(h, r) + η(h′, s).

17

18 CHAPTER 2. SUMMARY OF RESULTS

Theorem 2.1.4. Given two h-vectors h and h′ with a flats of degree d andd′ respectively. For the union we can then exclude h-vectors h′′ which havea flat of degree d′′ ≥ d, d′ with

max{η(h, d′′) + η(h′, d′′ + d′)− η(h′, d′) +

(d′ + 2

2

)− 2, η(h′, d′′)+

η(h, d′′ + d)− η(h, d) +(d+ 2

2

)− 2}< η(h′′, d′′) < η(h, d) + η(h′, d′).

Later, we used the concept of pseudo type vector (or partition) to givean algorithm that produces all the pseudo type vectors associated to a givenh-vector h, i.e., using this algorithm, we are able to find all monomial idealsI in two variables, x, y, with pure powers of the variables, such that the ringk[x, y]/I has h-vector h.Afterwards, we gave the definition of sum of two partitions, in such a waythat we could give a new algorithm, able to produce possible h-vectors ofthe union. We conjectured that this algorithm gives all of them.

In the last section, we provided a way to construct the unique minimum,with respect to the partial ordering defined on the set of h-vectors, h-vectorfor the union.

Proposition 2.1.5. For any two given h-vectors, we can always constructthe unique minimum h-vector for the union among all the admissible ones.This h-vector achieves the lower bound for the height and the upper boundfor the length.

The paper gives only a partial solution to the question, since we were notable to prove that the range of possible h-vectors found through the secondalgorithm covers all the possible h-vectors of the union, so we were not ableto give a complete characterization of the h-vectors of the union.

2.2 Paper B

Green’s Hyperplane Restriction Theorem: an extension to modules.

The second paper gives a new generalization of Green’s Hyperplane Re-striction Theorem to the case of modules over the polynomial ring.The tools we have used to prove the main theorem are mainly inequalities

2.2. PAPER B 19

regarding Macaulay representation of integers, lexicographic modules andgeneric initial modules. Namely, we have first proved the following inequal-ity:

Proposition 2.2.1. Given a, b ∈ N, a ≤ N1 =(n+d1−1

d1

), b ≤ N2 =

(n+d2−1d2

)and d1 ≥ d2, d1, d2 ∈ N, then

a〈d1〉 + b〈d2〉 ≤{

(a+ b)〈d2〉 if a+ b ≤ N2,

(a+ b−N2)〈d1〉 + (N2)〈d2〉 if a+ b ≥ N2.

Then, we have extended the inequality to a more general case.Finally, we have used the theory on generic initial modules to restrict to themonomial case and, combining the Green’s Hyperplane Restriction Theoremwith these inequalities on Macaulay representations, we were able to provethe main theorem.

Definition 2.2.2. LetM be a submodule in F , and m ∈ N. Set di = m−fi({d1, d2, . . . , dr} is a non-increasing sequence) and Ni =

(n+di−1di

)= dimkSdi

.Then, if

∑ri=j+1Ni ≤ H(F/M,m) ≤

∑ri=j Ni, for some j, we define

H(F/M,m){m,r} = (H(F/M,m)−r∑

i=j+1Ni)〈dj〉 +

r∑i=j+1

Ni〈di〉.

Theorem 2.2.3. Let F = Se1 ⊕ · · · ⊕ Ser where deg(ei) = fi for all i. LetM be a submodule in F , then

H((F/M)`,m) ≤ H((F/L)`,m)

where ` is generic linear form, m ∈ N, and L is a submodule that in degreem is generated by a lex-segment of length H(M,m).Moreover,

H((F/L)`,m) = H(F/M,m){m,r}.

In the past, a generalization of Hyperplane Restriction Theorem has beendone by Gasharov (see [Ga]). The reason why we did another generalizationwas to find a bound of the restriction of a module to a generic hyperplanefully described by lexicographic modules.

Green’s Hyperplane Restriction Theorem has been applied in severalpapers (see for instance [A-S], [B-Z], [M-Z]) to get some results about leveland Gorenstein algebras, with focus on the weak Lefschetz property. Forfuture work, we intend to use Theorem 2.2.3 to produce similar results.

References

[A-S] J. Ahn and Y. S. Shin, Artinian level algebras of codimension 3, J.Pure Appl. Algebra 216 (2012), no. 1, 95–107.

[B-Z] M. Boij and F. Zanello, Some algebraic consequences of Green’s Hy-perplane Restriction Theorems, J. Pure Appl. Algebra 214 (2010), no.7, 1263–1270.

[B-H] W. Bruns and J. Herzog, Cohen- Macaulay Rings, Cambridge Studiesin Advanced Mathematics, Cambridge University Press, 1998.

[Da] E.D. Davis, Complete Intersections of Codimension 2 in Pr: TheBezout-Jacobi-Segre Theorem Revisited., Rend. Sem. Mat.Univers. Po-litecn. Torino, 43, 4 (1985), 333-353.

[Ei] D. Eisenbud, Commutative algebra. With a view toward algebraic geom-etry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York,1995.

[Ei2] D. Eisenbud, The geometry of syzygies. A second course in commuta-tive algebra and algebraic geometry. Graduate Texts in Mathematics,229. Springer-Verlag, New York, 2005.

[Ga] V. Gasharov, Extremal properties of Hilbert functions, Illinois Journalof mathematics 41, 1997.

[G-M-S1] A.V. Geramita, J. Migliore and L. Sabourin, On the first infinites-imal neighborhood of a linear configuration of points in P2.J. Algebra 298 (2006), no. 2, 563-611.

[G-M-S2] O. Greco, M. Mateev and C. Söger, The h-vector of the unionof two sets of points in the projective plane. Matematiche (Catania) 67(2012), no. 1, 197–222.

21

22 REFERENCES

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[Gr2] M. Green, Generic initial ideals, Six lectures on commutative algebra,119–186, Mod. Birkhäuser Class., Birkhäuser Verlag, Basel, 2010.

[H-H] J. Herzog and T. Hibi, Monomial ideals. Graduate Texts in Mathe-matics, 260. Springer-Verlag London, Ltd., London, 2011.

[Hu] H. Hulett, A generalization of Macaulay’s theorem, Communicationsin Algebra 23 (1995), 1249-1263.

[Ka] G. Katona, A theorem for finite sets, Theory of graphs, AcademicPress, New York, 1968, 187-207.

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[Kr] J.B. Kruskal, The number of simplices in a complex, Mathematicaloptimization techniques, Univ. of California Press, Berkeley, 1963, 251-278.

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