Annali di Matematica pura ed applicata (IV), Vol. CLXXIX (2001), pp. 295-307

Linear Systems, Singularity and Not Too Ramified Morphisms (*).

M&RIA LUISA SPREAFICO

Abstract. - Given a smooth k-variety Y (where k is a field of arbitrary characteristic) and a linear system on Y we study the dimension of the singular locus of the general element of ~, both inside and out-

side the base locus B of or We interpret these results from the point of view of the transversality the- ory, and we improve a result by Speiser about the not too ramified morphisms. Moreover, we show that our results can be applied in some cases where a criterion by Zhan~ for the smoothness of the general element of ~, fails.

O. - Introduction.

In this paper we want to study some problems strictly connected with the transversality theory (see, for instance [4]).

To explain our results we recall the main problem of the transversality theory. We con- sider the fibred product diagram:

W p 2 y

(*) X ~, Z

S

where X , Y , Z and S are smooth varieties. Given a pair ( f , ~) of ~good~ morphisms and a morphism g which ranges in a fixed class

of morphisms, the classical problem of transversality is to give a criterion for the general fiber W, =/)1-1 (~-1 (s)), with s ~ S, to be smooth (i.e. W, is smooth when s varies in a non empty open subset of S).

Many results were given by Kleiman [4], Laksov and Speiser [5~ 6, 8] when g ranges in the family of closed immersions. Other properties than smoothness were studied by Zhang [11] for morphisms g ranging in the family of closed immersions, and an axiomatic approach

(*) Entrata in Redazione il 15 settembre 1997, e in versione definitiva il 28 ottobre 1999. Indirizzo dell'A.: Dip. Matematica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129

Torino, Italy.

296 MARIA LUISA SPREAFICO: Linear systems, singularity, etc.

was given by Cumino, Greco, Manaresi [1], and myself [9, 10] for morphisms g belonging to the bigger family of residually separable morphisms.

Another particular class of morphisms g, the not too ramified morphisms, was consid- ered by Speiser [8]. Because we are interested in such a class, we recall the definition of not too ramified morphism.

DEFINITION 0.1. -- Let Y and Z be two smooth varieties. A morphism g : Y---~ Z is not too ramified if the corresponding tangent morphism Tyg:TyY---~Tg(y)Z between the tangent spaces Ty Y and Tg(y) Z of y and g(y) respectively, is not the zero map for a dense set of y ~ Y .

For these morphisms Speiser proved that the general fiber Ws is generically smooth (see [8, Prop. 2.1]; <<goo&~ conditions on the pair (f, Jr) are assumed).

Our aim is to give an upper bound for the dimension of the singular locus of the general fiber W,. Moreover, we give a result which works for all the morphisms but which is more interesting for the not too ramified ones.

We study a particular but fundamental diagram (*) that reminds the Classical Bertini Theorem. In fact, in diagram (*) we set: Z = pr, S = (P')*, X the incidence relation point- hyperplane, while f and z~ are the projections onto pr and (pr)*, respectively. We call (**) the diagram obtained with these assumptions.

For this specialization of the diagram, the general fiber W, represents g - I (H) for the general hyperplane H of p r.

Our approach to the problem consists in studying the singular locus of the general ele- ment of a linear system 8. Then, we interpret g - 1 (H) as an element of the linear system s on Y associated to g, in the canonical way. In this way we obtain a bound for the dimension of the singular locus of the general fiber W,.

Moreover, we extend the results to any variety Y not necessarily smooth. In this case we need a generalization of the definition of not too ramified morphism when the domain is a singular variety (see Def. 3.6). Also in this general assumption we give an upper bound for the dimension of the singular locus of the general Ws.

The organization of the exposition is the following. In Section 1 we fix some conventions and notation. In Section 2, we give a bound for the dimension of the singular locus of the general ele-

ment of a linear system 8 both inside and outside the base locus B of 8. The method is based on the computation of two sequences: (i) a sequence (r0, ..., rl) of integers associated to a fi- nite number of Jacobian matrices and (ii) a sequence (e0 . . . . . Cl) of dosed subvarieties de- fined using Jacobian matrices and the first sequence. We link these sequences with a given number ~e~ introduced by Greco and Valabrega [2] to obtain the required bound. The computation of these sequences can be performed using any computer algebra system, like CoCoA, Macaulay and Reduce.

At the end of the section, we develop several examples and applications where our bound is sharp.

In Section 3, we define an integer ~<h~ that gives the ramification ~type)) for a morphism

Mt~aA LUISA SP~AFICO: Linear systems, singularity, etc. 297

g and we give a bound for the dimension of the singular locus of the general fiber Ws in the diagram (**). In particular, when g is not too ramified (i.e. h < dim (Y) - 1 ), we improve [8, Prop. 2.1].

Moreover, we show that our results assure the regularity for the general fiber Ws, in some cases where a criterion by Zhang fails [see Remark 3.4].

Finally, we extend the results for the diagrams (**) with Y singular. We wish to thank Prof. S.Greco for a useful conversation on the subject of this

paper.

1. - C o n v e n t i o n s and Notation.

In this paper a variety is the ringed space of the'closed.points of a k-scheme of finite type, where k is an algebraically closed field of characteristic p I> 0.

If V is a variety, then we write Wc V to mean that W is a locally closed subvariety of V. We consider the empty subvariety as having dimension - 1 .

Now, let Y be a smooth variety of pure dimension n. We denote by ~3r the tangent sheaf (or sheaf of derivations) which is a locally free sheaf

of rank n -- dim (Y) (cfr. [3, Ch.II, 8]). Let ,l~ be an invertible sheaf on Y and let 8 _c H 0(y, ,1~) be a linear system generated by

global sections corresponding to the (Cartier) divisors F0, . . . , F,. We can identify 8 with p r writing Lx = X i2 iF,, for each ;t = (~. 0, . . . , 2 r) e P ' . For every open subvariety "Lt of Y we shall use local equations f0 . . . . . f, of the divisors Fo . . . . . Fr, and it is understood that these equations are taken in a compatible way.

For later use, if B is the base subvariety associated to 8, we define C = e \ B , for each subvariety C of Y.

Finally, we say that any property 8' holds for the general Lx if the set {2 e pr I 8' holds for the corresponding Lx } contains a non empty open subset of P ' .

2. - Linear Sys t e ms on a project ive variety Y.

From now on, we assume that Y is an integral smooth variety of dimension n (see Remark 2.10 for a generalization if Y is singular).

If 8 is a linear system on Y, we want to give a bound for the dimension of the singular lo- cus of the general element of 8 both outside and inside the base locus B. For brief, we write these dimensions as dim (Sing (Lx) N ~ and dim (Sing (Lx) N B), respectively. In particular, the bound about the first dimension allows us to define the type of a ramified morphism in the next section.

To this aim we shall use the bound [2, Th. 2.3] that involves the integer ~<e(y))) defined as [2, Def. 2.1]:

DEnNITION 2.1. -- Let y ~ Y and let e be an integer. We say that 8 verifies Te at y i f there are e divisors L1, . . . , Le whose local equations at y are linearly independent modulo 8E~ (where

298 MARIA LUISA SPREAFICO: Linear systems, singularity, etc.

BEy is the maximal ideal of Oy, y). We denote by e(y) the maximum integer e for which T~ is verified at y.

(2.a) How to compute e(y).

Whenever we fix a point y e Y, we can consider all the divisors of 8 passing through y, that is, the linear subsystem of 8 generated by the divisors L[ . . . . . L~(y), where ~(y) = r (re- sp. r + l ) i f y~ tB (resp. yeB) .

Let Iy = ( f ; . . . . , f~(y))_c Oy. r be the ideal generated by the equations f ' of the divisors Li', for i = 1, . . . , Q(y).

I f we choose D1 . . . . . Ds e ~y, y, then we write J(Iy; Da, ..., 1),) = (Dif} mod 8Ey). This is a ~) x s matrix with entries in the field �9 r/SKy.

Under the above hypotheses and notation, we can prove the following proposition which is essentially [7, Th. 30.4].

PROPOSmON 2.2. -- Let be r(y) := max {s[ there exist s derivations D1 . . . . . 1:),, such that rkJ(Iy; D1 . . . . . D,) = s}. Then we have: e(y) >I r(y).

In particular, there exist r( y ) divisors, suppose L[ . . . . , Lr'(y) , such that dimy (LI' fq ... A Lr'(y)) = n - r(y) and y E Reg (El' N ... fq Lr'(y)).

PROOF. -- We can always suppose that det (Di f / ) ~t 8Ey, for r(y) >~j, i/> 1. Then we have that the images in OEy/OE~ o f f [ . . . . ,if(y) are linearly independent over �9 y/OEy.

The inequality e(y) >I r(y) follows by Def. 2.1. �9

REMARK 2.3. -- If r(y) = 0 for some y e Y, then for each divisor L such that y E L we have that y ~ Sing (L). In fact, if y e L = 3.1L1' + ... + 3. rLr ' for suitable 3.i 's, then y satisfies the equation of L and of all derivations Dif/(y) -- 0, whatever 2 i.

COROLLARY 2.4. -- Under the above hypotheses, for each y ~ Y:

(i) e(y) = r(y) + 1, i f yE Y;

(ii) e(y )= r(y), i f y ~ B .

PROOF. - - Applying [2, Ex. 2.2(i)], we have that 8 verifies Te at y if and only if either of the following conditions is satisfied: a) there are L1 . . . . . Le~ 8 such that dimy (Lt N ... N Le) = = n - e and y eReg (L1N ... n Le); b) there are L1, ..., L~-I as in a) and moreover there is L~ e 8 such that y ~ Le. Then the equalities follow from Prop. 2.2. �9

(2.b) A stratification of Y and the required bound.

In order to study dim (Sing (Lx) • ~ and dim (Sing (L~) N B), we want to give a stratifi- cation for both ~" and B. In the following, we write Co for ~" or B without distinction, to de- scribe the two stratifications at the same time.

We remark that the number r(y) ranges over a finite number of integers, whenever y e Y, because n -- rk~r>~ r(y) >- O.

MARIA LUISA SPREAFICO: Linear systems, singularity, etc. 299

Then we have a numerical sequence of admissible ranks:

(2.5) rk(73r, 8, C o ) = ( r o , . . . , r l ) with r o > r l > . . . > r l

where ri is an admissible rank if there exist at least a point y ~ Y such that ri = r(y). We can associate to each integer ri a closed subvariety of Y using the following:

PROPOSITION 2.6. -- Let ri be an admissible rank. Then, the set Ci+ 1 of the closed points y ~ Co such that r(y) < ri, is a closed subvariety of Y (with the induced reduced struc- ture).

PROOF. -- We can construct a finite open affine covering U = {~s},= 1 . . . . . . of Y which satisfies the two conditions: (i) 73r/~, is generated by the global sections D1 ~ . . . . . D,~, and (ii) the linear system 8 is generated by the divisors F0, . . . , F, with local equations f~, . . . , f i in "It,.

We show that the set G + 1 is an algebraic set in each open subset "1~,. When "1/, f) ~ t # 0, we glue the local equations using the restriction maps both of the sheaf 13y and of the linear

system 8.

Case: Co = ~". We fix the open set "tl, and we suppress the index s in the following. We need to consider all the divisors passing through a given point y ~ Y. Replacing 'L/with r ~'~ ~fr ' if necessary, for each y e ~" we can suppose

zy= (fo_ fo(y ). f,-l(y) f, I f'-~7) fr' " ' ' ' fr-1 f-~-~y) " ]"

Then, the r x n matrix

J(y) = [ D h f i ( y ) - f-~y) ]

is the matrix J(Iy; D1, ... , Dn), whenever y ranges in ~ .

Case: Co = B. It is similar to the previous case, but we consider B A ~ and the idea] Iy =

= (f0 . . . . . s because y e B . �9

Now, we can consider the subvariety Yi (resp. the dosed subvariety Ci+ 1) of the points y e e0 such that r(y) = ri, (resp. r(y) < ri). Also the subvarieties Yi are considered with the reduced induced structure and we have the inclusions Yic_ Ci, for i = O, . . . , l, (with

YI = el). We remark that {Yi} is a stratification of Co. Considering the sequence of dosed subvarieties (Co, . . . , el), we can define the numeri-

ca] sequence of the associated dimensions

(2.7) dim (~r, 8, Co) = (dim (Co), . . . , dim (Ct))

300 MAmA LuIsA SP~AFICO: Linear systems, singularity, etc.

PROPOSITION 2.8. -- Under the above assumptions, let Lx be the general element o f the lin- ear system 8 on Y . Then:

(i) dim (Sing (L~)n ~ ~< max { dim ( C i ) - r ~ - 1]i = 0, . . . , l};

(ii) dim (Sing(L~) AB) ~ max {dim (Ci) - ri I i = 0 . . . . , l}.

PROOF. -- Case: Co = ~'. For each i = 0, . . . , l, the locally closed subschemes Yi can be de- composed in irreducible components. Let Yi be the union of the components of maximal di- mension. (We consider these particular components because they give the worst case for the bound).

Applying [2, Th. 2.3] and Corollary 2.4(i), we have that d im(YinS ing(L~) ) , , < ~< dim (Yi) - ri - 1.

Recalling that dim(Yi) ~<dim(Ci) we obtain: dim(Sing(La)) =d im( (UYi ) A n Sing (L~)) = max { dim ( Yi n Sing (Lx)) [i = 0, . . . , l} ~< max { dim (e i) - r i - 1 ]i = O, . . . , l}.

Case: Co = B. In this case, the proof is similar but we apply Corollary 2.4(ii) at the first step. �9

RuMaV4: 2.9. - Let us observe that, if k is a field of characteristic zero, the Classical Bertini Theorem assures that the general element L~ of the linear system 8 on Y is smooth outside the base locus. In this case the interesting bound is given by (ii) of the above proposition (see also Example 2).

REMARK 2.10 (Y singular). - We can generalize Proposition 2.8 to any reduced variety Y of pure dimension n, not necessarily smooth and irreducible.

In fact, we consider the closed subvariety F i of Y defined as Fj := Sing (F i_ 1 ) with F0 := = Y , with the induced reduced structure. We define the smooth subvariety Gj = Fj \Fi+ 1 of Y for j = 0 . . . . . N, and we remark that Y-- UiG j.

For s -- 1 . . . . , m(j ) , let Gjs be the irreducible components of Gj, considered with their reduced structure. For each Gss it is possible to construct the stratification described in sub- section (2.b) finding the two fundamental sequences of closed subschemes and ranks:

(e~', . . . , C[(js)) and (rg . . . . . J' rt~)).

Now, we can apply Proposition 2.8 to each irreducible component Gj, obtaining, for the general L~

dim (Sing (La) n G~) ~ his = max {dim (C] ~) - r p - 1 [ i = o, . . . , l ( j s ) }

and hence:

dim (Sing (Lx)n Gj) ~< h i= max {his I s = 1, . . . , m(j)} .

MARIA LUISA SPREAFICO: Linear systems, singularity, etc. 301

Finally, gluing together all the Gj, we obtain

dim (Sing (Lx) A ~ ~< h = max {bj [j = 0 . . . . . N} .

Using the same procedure, we can deduce a bound for dim(Sing(L z) A B).

(2.c) Examples.

The following examples will remark that:

(1) the last admissible rank rt can be different from zero (some cases of Ex. 2);

(2) it is possible that d i m ( e / ) = d i m ( C i _ l ) although Ci ;~ Ci_ 1 for some index i (Ex. 1, and some cases of Ex. 2);

(3) also if 8 and 8' are two linear systems on Y of the same dimension with dim (B) = = dim (B ') and dim (Sing (La)) = dim (Sing (L~')), it can happen that the two respective fun- damental sequences are different (Ex. 1);

(4) the bounds are sharp in several examples (Ex. 1, and some cases of Ex. 2).

All the examples concern linear systems 8 on Y = p4. With the notation of Section 1, we suppose that the elements of 8 are of the form Lx = 2~i2 iFi, for i = 0, . . . , r. We stratify Y = = p 4 , as showed in Subsection (2.b), choosing the canonical affine covering U = = { As }s = 0 ..... 4. We observe that the generators of ~WA, are the well known partial derivatives Dj = 3/3yj for j = 0, . . . , 4, j ~ s, and that the divisors Fi admits global homogeneous equa- tions fi, in the variables Yo, . . . , Y4.

Instead of computing the five matrices ]0 . . . . . ]4, corresponding to the fundamental affine open subsets of the projective space, we shall write only the matrix J = [Sfi/Sy i] for i = 0, . . . , r and j = 0, . . . , 4, that contains all the derivatives with respect to the homoge- neous variables y0 . . . . , Y4. In fact, we obtain ]i as a submatrix of J by erasing the i-th column and setting Yi = 1 in the other columns.

Then, we compute the two fundamental sequences rk (~p4, 8, C0) and dim (~p4, 8, CO) and we apply the bounds given in Prop. 2.8.

To check our bound, we start each example giving the dimensions dim (Sing (Lz) N ~:~) and dim (Sing (Lz)fq B) for the general ;L, which are computed by solving the parametric system

{ SLa/Sy~ = 0 0 <~ j <. 4

(P) L~ = 0

b i = 0 O<~i<~s

where I = (b0, . . . , bs) is the reduced ideal defining the base locus B set theoretically (we consider the last equations only to compute dim (Sing (Lz) A B)).

Let us observe that it is easier to perform our computation than to solve the parametric system (P).

We recall that p denote the characteristic of the ground field k.

302 MaRia LuisA Sv~AFICO: Linear systems, singularity, etc.

EXAMPLE 1. -- Let 8 be the linear system of p4 generated by the divisors F0, . . . , F4 of equations

9Co := Y~, fl := Yl D, f2 := Yf, f3 := Y~ Y~ - 2, f4 := Y~ + y~e + yo 2 y3e -2

It is easy to show that B = 0.

Case p >>- 3.

By solving the system (P), we have that dim (Sing (La)) = 2, for the general ft. To compute the two fundamental sequences we need the matrix J:

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 2y3y4e -2 -2y~y,~ -3

2yoy~-2 0 0 - 2 y ~ y ~ - 3 0

We have that: rk (~p4, 8, p4) = (2, 1, 0) and dim ('~p4, 8, p4) = (4, 3, 3 ). In fact, G3 = p4 and, set theoretically, Cl contains the hyperplane y0 = 0 while e2 contains the hyperplane Y3 = 0.

By applying our bound (Prop.2.8 (i)), we obtain that dim(Sing(L~))~<2, that is sharp.

Case p = 2.

In this case dim (Sing (La)) = 3 for the general ft; in fact, each hypersurface is double. Moreover, the matrix J is the zero matrix and we obtain the two trivial sequences: rk(~p, , 8, p4) = (0) and dim Ugp4, 8, p4) = (4).

Our bound (Prop. 2.8(i)) gives dim(Sing(Lx))~< 3, (sharp). Now we consider the linear system 8' generated by the divisors of equations

go :=YS, gl :=Y~, g2 :=Y:r g3 :=y~+yf fy~-2 , g4 :=Y~.

Similarly to the linear system 8, we have that B ' = 0 and dim (Sing (Da')) = 2 if p I> 3, while dim (Sing (La)) = 3 if p = 2, for the general ft. But we have different fundamental se- quences. In fact, for p~>3, we obtain: rk(~gv4, 8', p 4 ) = (1, 0) and dim Ugp4, 8', p 4 ) = = (4, 3). While, for p = 2 , we obtain rk(~p4, 8', p4) = (0) and dim (~6~4, 8', p4) = (4).

Let us observe that our bound is sharp in these cases, too. In the following example we give a linear system of degree n and we analyze the most im-

portant values of n and p. The other cases are left to the reader. We recall that, in characteristic zero, the only interesting bound involves the dimension

of the singularity inside the base locus B (see also Remark 2.9).

MARIA LUISA SPREAFICO: Linear systems, singularity, etc. 303

EXAMI'LE 2. -- Let 8 be the linear system of p4 generated by the divisors F0 . . . . , F 4 of equations

fo : = Y ~ - Y l Y 2 Y ; -2 , f l :=y2y•-2, f2 : = Y ; , f3 : = y f - 2 y 2 , f4 :=y~-2y2 .

The base locus B is non-empty and dim (B) = 2 if n ~e 2, or dim (B) = 0 if n = 2, with defin- ing ideals I = (Y0, Y3) and I = (Yo, yl, Y3, Y4), respectively.

C a s e p = n = 2 .

By solving the parametric systems (P), we have that dim(Sing(L.~)NP 4) = 1, and dim (Sing (L~) n B) = - 1, for the general 2.

Using the matrix ] associated to 8 we have the two pairs of sequences:

rk ("l~p4, 8, ~4) -- (1, 0) ; dim (~p4, 8, ~:~4) = (4, 2)

rk ("~p4, 8, B) --- (1 ) ; dim (~p4, 8, B) = ( 0 ) .

By applying our bounds (Prop. 2.8) we obtain that dim (Sing (La) N ~4) _< 2 (not sharp), and dim (Sing (La) n B) ~< - 1, (sharp).

Case p = n = 3.

By solving the parametric systems (P), we have that d im(S ing (L~)AP 4) - -0 , and dim (Sing (L~) N B) = 1, for the general 2.

Using the matrix J associated to 8 we have the two pairs of sequences:

rk(~p4, 8, P4) = ( 3 , 2 , 1, 0) ; dim(~p4, 8, P4) = (4, 3 , 3 , 1)

rk('~p4, 8, B) = (1, 0) ; dim (~p4, 8, B) - (2, 0) .

By applying our bounds (Prop. 2.8) we obtain that dim (Sing (L~) n ~4) ~< 1 (not sharp), and dim(Sing (L~) N B) ~ 1 (sharp).

Case p ;~ 2 , n = hp >~ 5 , h ~ N .

By solving the parametric systems (P), we have that d i m ( S i n g ( L x ) N P 4) = 0 , and dim (Sing (L~) n B) = 2, for the general A.

Using the matrix J associated to 8 we have the two pairs of sequences:

rk(~v4, 8, P4) --- (3, 2, 1, 0) ; dim(~v4, 8, P4) - (4, 3, 3 , 3 )

rk(~p4, 8, B) = (0) ; dim (~p4, 8, B) = (2) .

By applying our bounds (Prop. 2.8) we obtain dim (Sing (L~) N ~4) ~< 2 (not sharp), and dim (Sing (L~) N B) ~< 2, (sharp).

Case p # 2 , n ~ hp , n - 2 ;~ hp , h ~ N .

304 MARIA LUISA SPREAFICO: Linear systems, singularity, etc.

By solving the parametric system (P), we have that dim(Sing(La)n ~ 4 ) = - 1 and dim (Sing (La) n B) = 2, for the general 2.

Using the matrix J associated to 8 we have the two pairs of sequences:

rk(~p4, 8, PP4) = (4, 3 ,2 , 1); dim(~p4, 8, P4) = (4, 3 ,3 , 3).

rk(~v4, 8, B) = (0); dim (~6p4, 8, B) = (2).

By applying our bounds (Prop. 2.8) we obtain dim (Sing (La) n ~4) ~ 1 (not sharp), and dim (Sing (La) M B) ~ 2, (sharp).

Case p =0 , n -- 2.

By solving the parametric system (P), we have that dim (Sing (La) n B) = - 1 , for the general 2.

Using the matrix J associated to 8 we have the pair of sequences

rk(78p4,8, B ) = ( 1 ) ; d im(~p4,8 , B ) = ( O ) .

By applying our bound (Prop. 2.8(ii)) we obtain: dim (Sing(La) n B) ~< - 1 (sharp).

Case p =0 , n =3.

By solving the system (P), we have that dim (Sing (La) n B) = 1, for the general 2. Using the matrix J associated to 8 we have the pair of sequences:

rk(78p4,8, B ) = ( 1 , 0 ) ; dim (~p4, 8, B) = (2, 0).

By applying our bound (Prop. 2.8(ii)) we obtain dim (Sing(La) N B) ~< 1 (sharp).

3. - N o t too ramified morphisms.

Let us consider the fibred product diagram:

W p2 y

(**) X s p ,

(pr)

where Y is an integral variety of dimension n, X is the incidence relation point-hyperplane and the morphisms f and x are the projections onto pr and (pr)* respectively.

(3.a) First Case: Y is an integral smooth variety.

Let g : Y--+ P ' be a morphism. We can associate to g the invertible sheaf ,eg = g * ((9( 1 )) generated by the global sections So, . . . , s r e H ~ 2~g).

MARIA LUISA SPREAFICO: Linear systems, singularity, etc. 305

These sections define the Cartier divisors Fo, . . . , F, in a canonical way and we write s for the linear system generated by these divisors. We observe that the linear system s is base point free.

We can compute the two fundamental numerical sequences (2.5) and (2.7), as explained in Section 2:

r k ( ~ r , s Y) = (r0, . . . , rl) and d im(~v , s Y ) = (dim(C0), . . . , dim(C1)).

We remark that these two fundamental sequences are obtained from the matrices J(Iy; DI, . . . , Ds) which describe the tangent morphisms Tyg. If r0 ~ 0 then the morphism g is not too ramified in accordance with Def. 0.1 (see also [8]).

Now, the idea is to consider all the possible ranks (ro . . . . . rl) and to give a <<stratifica- tion>~ of Definition 0.1 (not too ramified morphisms). Our Definition 3.2 is suggested by Proposition 3.1 and it is really interesting for the not too ramified morphisms (it is trivial when r0, and then all the ri's, are zero).

PROPOSITION 3.1. -- Let us consider the fibred product diagram (**). Then there exists a dense open subset ~0 of (pr)* such that for all s e ~0 we have:

dim (Sing (W,)) ~ h :--- max {dim (Ci) - ri - 1 l i-- o, . . . , l}

PROOf. - The claim is an easy consequence of Prop. 2.8. �9

From the previous proposition, we can define the <<type>~ of a morphism in a natural way.

DEmNmON 3.2. -- Let Y be a smooth variety and let g: y__>pr be a morphism, with associated fundamental sequences rk (~v, ~g, Y) = (ro, . . . , rl) and dim (~r , s Y) = = (dim (~) , . . . , dim (el)).

We say that g is ramified of type h i f h = max {dim (el) - ri - 1 l i = 0, . . . , l}.

Rewriting Proposition 3.1, we obtain the following result which can be compared with [8, Prop. 2.1].

COROLLARY 3.3. -- Let us consider the fibred product diagram (**) where g is a ramified morphism of type h. Then, for the general fiber Ws, we have that

dim (Sing (Ws)) <~ h .

We remark that: (i) if g is not ramified, then ri = dim Y for each i, and we have h = - 1. (ii) If g is not too ramified then we have h < dim Y - 1 ; in particular the general fiber is generically smooth and Coroll. 3.3 improves [8, Prop. 2.1].

REMARK 3.4. -- Zhang [10, Th. 2.1] gives a criterion for the regularity of g - I ( H ) for the general hyperplane H of P ' , or equivalently, for the general fiber Ws.

To apply that criterion it is necessary to find a stratification of Y with smooth k-sub-

306 MARIA LUISA SVREAFICO: Linear systems, singularity, etc.

schemes Ri, i= 1 . . . . , m, such that, for each i, the morphism g/Ri: Ri-+g(Ri) is smooth (where ~(Ri) is the closed subscheme induced by g(Ri) with the reduced structure).

In the following example we give a morphism g : A 2 --+ p3, where the ground field k is of positive characteristic p > 0 and we show that Zhang's stratification doesn't exist, but it is possible to apply Prop. 3.1 to verify that g - 1 (H) is smooth for the general hyperplane H of p3.

Let g : Ae--+A3--+P 3 be the morphism defined as g(u, v) = (u : vP: uv: 1). We can use only one matrix J to describe the numbers r(y), VyeAZ:

1 0

:1 We have the two fundamental sequences: rk (~a2, Ag, A 2) = (2, 1) and dimU$A2, Ag, A 2) = (2, 1 ); then, applying our bound (Prop. 3.1), we obtain dim (Sing (Ws)) ~ - 1, i.e. the fibre g - I ( H ) is smooth for the general hyperplane H of p3.

Zhang's stratification doesn't exist. In fact one of the subschemes Ri, suppose R1, must contain a dense open subset of A 2 such that g& is smooth. Some other R s, suppose R2, must contain an open dense subset of the closed subscheme of A 2 given by the equation u = 0. But g/R~ is not smooth.

(3.b) General Case: Y is an integral variety.

We can generalize Definitions 0.1 and 3.2 choosing the integral variety Y not necessarily smooth.

In fact, we can stratify such a variety Y, as explained in Remark 2.10, using the smooth locally closed subvarieties Gj = Fj\Fj+ 1 = Fj\Sing (F i) for j = 0, . . . , N, with F0 = Y.

Using this stratification we generalize Def. 0.1:

DEVlNmON 3.5. -- Let Y be a variety and let Z be a smooth variety. For all j = 0 . . . . , N, let us consider the subvarieties Gy of Y and a point y ~ Gj.

A morphism g : Y-->Z is not too ramified i f for each j, the corresponding tangent mor- phism Ty g : Ty Gj--+ Tg(y) Z between the tangent spaces Ty Gj and Tg(y) Z of y and g(y) respect- ively, is not the zero map for a dense set of y ~ Gj.

Using the notation of Remark 2.10, we can associate to any morphism g all the funda- mental sequences that generalize the corresponding sequences of Def. 3.2:

rk (~c,~, ~g, Gjs) = (rfl) and dim (~cj,, ~g, Gj~) = (dim (C[s))

for j = O, . . . , N, s = 1, . . . , m(j) and i-- 0, .. . , l(js).

DEFINITION 3.6. -- With the above notation, we say that g is ramified of type h i f h = max{dim (e js) - r/~ - 1 }.

l , J , S

MARIA Ltnsa SPr, EAF~CO: Linear systems, singularity, etc. 307

W e observe two facts: (i) if r6 s # 0 for aU j , s, then g is not too ramified in accordance with Def. 3.5; (ii) whenever Y is smooth, then Fj -- F0 = Y for each j and we obtain again Definition 3.2.

Now we can generalize Proposi t ion 3.1:

PROPOSITION 3.7. -- Let us consider the fibred product diagram (**) where Y is an integral variety, not necessarily smooth, and g is a ramified morphism of type h.

Then there exists a dense open subset ~0 of ( P ' ) * such that for all s E ~ we have:

dim (Sing (Ws)) ~ h .

N

PROO;. - For a fixed s ~ S we have that W~ -- U (W~ x yG~); in a geometrical way we can j = 0

write W~ = U i ( W ~ N Gi). Then, applying Prop. 2.8 to each irreducible componen t of Gj, we can say that there exists a dense open subset ~ of S such that dim (Sing (W~) N Gj) ~ h i, for each s ~ ~ .

Then, for each s e V l j ~ = ~9, we obtain dim (Sing (Ws)) ~< h = max {hi}, gluing together the previous inequalities, i

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