On the Bertini theorem in arbitrary characteristic

We give a simple direct proof of the Kleiman Bertini theorem in arbitrary characteristic. We also give a simple proof of Serre splitting theorem.


Introduction
Kleiman in the paper [4] proved Bertini theorem in arbitrary characteristic (Corollary 12 in [4]). Kleiman deduced this theorem from his general transversality theorem: if X is G-homogenous algebraic variety and Z , W are smooth varieties over X , then the fiber product of Z and a general translate gY over X is smooth. This theorem is valid in characteristic zero, however Kleiman shows that his result still works in positive characteristic, if we additionally assume that the action of the group G is sufficiently good (this assumption is satisfied for the action of the group PG L(n) on the projective space P n ) and that Z , W are unramified over X.
The aim of this note is to give a simple, direct proof of the Bertini Theorem in arbitrary characteristic. We also partially generalize our method to locally free sheafs of higher ranks (mainly for char k = 0). In particular we give a simple proof of the fact that if a variety X is smooth and F is a locally free sheaf on X , with sufficiently many sections, then a generic section of F is transversal to X. As a Corollary we give a proof of the Atiyah-Serre Splitting Theorem (compare with [6], [1] and [5]).

Notations and definitions
We assume for simplicity, that the base field k is algebraically closed. Let X be algebraic variety and F be a locally free sheaf on X of rank r. Take a section s ∈ (X, F). We describe a scheme of zeroes of s −1 (0) in the following way: locally we can assume that X = U is an affine variety and We say that the section s is transversal to X if either s −1 (0) is smooth and it has dimension dim X − r or s −1 (0) = ∅.
Let X, Y be smooth varieties and f : X → Y be a morphism. We say that f is unramified, if f separates infinitely near points of X , i.e., the mapping d x f : T x X → T f (x) Y is a monomorphism for every closed point x ∈ X (see [3], p. 15).
If X is an affine variety we will denote by k[X ] the ring (X, O X ). If M is a k[X ] module then by M ∼ we denote the sheafification of M-see [2], Definition on the page 110.

Main result
Theorem 3.1 (Bertini theorem in arbitrary characteristic) Let X be a smooth algebraic variety of dimension d and let F be an invertible sheaf on X . Assume that F is generated by global sections s 1 , . . . , s r ∈ (X, F). Assume that the morphism from X to the projective space P r −1 given by s 1 , . . . , s r is unramified. Then there is a Zariski open non-empty subset U ⊂ k r such that for every c = (c 1 , . . . , c r ) ∈ U the zero set of the section s = r i=1 c i s i is smooth. Proof Since X is quasi-compact we can assume that X is affine and the sheaf F is trivial. Hence we can identify F with O X and now s i : X → k are regular functions. Additionally we can assume that s r ≡ 1. Indeed, since s 1 , . . . , s r generates F we have that open (affine!) subsets U i := X \{s i = 0} cover X. Consequently we can assume that s r = 0 in X. Take s i = s i /s r , i = 1, . . . , r. If we prove our theorem for s i , then we automatically prove it also for originals s i . Indeed, note that for a . Moreover, we can assume that on X we have global local coordinates The variety V is smooth. Indeed, let (g 1 , . . . , g r ) be the set of generators of the ideal Consider the projection: We show that for generic c ∈ k r the mapping q is smooth on the set Indeed, let us compute the tangent space at (c, x). It is given by the equation Let us note that rank Let us note that the mapping q is not a submersion in a neighborhood of q −1 (c) a exactly if L c = ∅. We show that the set S := {c ∈ k r : L c = ∅} is a constructible subset of k r of dimension less than r.
Let us note that the fiber of π is a linear subspace of k r −1 of dimension r − 1 − d ( as a kernel of a suitable linear mapping). Hence dim W ≤ r − 1. Let We have a surjective mapping This means that the projection q : V → k r is a submersion outside a proper algebraic subset S ⊂ k r . In particular the zero set of a generic section s = r i=1 c i s i is smooth.
Using a similar method we can generalize this result to higher dimension. As a Corollary we obtain Atiyah-Serre Spliting Theorem. F is generated by global sections s 1 , . . . , s r ∈ (X, F). Then there is a Zariski open non-empty subset U ⊂ k r such that for every c = (c 1 , . . . , c r ) ∈ U the zero set of the section s = r i=1 c i s i is either empty or it has dimension d − n. Moreover, if char k = 0 and X is smooth, then a generic section s = r i=1 c i s i is transversal to X. Proof Since X is quasi-compact we can assume that X is affine and the sheaf F is trivial, i.e., we can identify F with O n X . Moreover we can assume that on X we have a global local coordinates (x 1 , . . . , x d ). Hence s i = (s i1 , . . . , s i,n ) : X → k n are regular mappings. Let

Theorem 3.2 Let X be an algebraic variety of dimension d and let F be a locallyfree sheaf on X of rank n. Assume that
Let us note that dim V = r +d −n, where d = dim X. Indeed, we have a surjection π : V (c, x) → x ∈ X. Any fiber of π is a linear subspace of k r of dimension r − n ( it is a kernel of surjective linear mapping F x : k r c → r i=1 c i s i (x) ∈ E n x , where E n = X × k n is a trivial vector bundle of rank n). Consequently dim V = dim X + r − n = d + r − n. Now consider the second projection: If it is dominated then the generic fiber has dimension dim V −dim k r = d +r −n−r = d − n, otherwise the generic fiber is empty.
Moreover, if X is smooth over k, then V is smooth. Indeed, let (g 1 , . . . , g r ) be the set of generators of the ideal . . . , c r , x 1 , . . . , x d ) is a set of global local coordinates on k r × X. Let us note that polynomials h j = r i=1 c i s i j (x) does belong to the ideal I (V ). Now we see that rank J (V ) ≥ n, because partial derivatives of h j with respect to c i form a matrix [s i j (x)] 1≤i≤r,1≤ j≤n , which has a rank n. Hence dim T (c,x) V ≤ d +r −n. On the other hand dim V = d + r − n and consequently we have the equality dim T (c,x) V = d + r − n = dim V. If additionally char k = 0, then the generic fiber is also smooth (generic smoothness-see [2], Corollary 10.7, p. 272). Corollary 3.3 (Atiyah-Serre Splitting Theorem) Let X be an algebraic variety of dimension d and let F be a locally free sheaf on X of rank n. Assume that F is generated by global sections s 1 , . . . , s r ∈ (X, F). If n > d then F contains a trivial subsheaf A ⊂ F of rank n − d.
Proof Indeed, by induction we can find n − d linearly independent sections t j = r i=1 c j,i s i (note that the quotient sheaf of F is generated by the same sections (or rather their classes) as F).
In particular if we assume that X is affine we have the Serre result in a classical form: Corollary 3.4 (Serre splitting theorem) Let X be an affine algebraic variety of dimension d and let F be a locally free sheaf on X of rank n. If n > d then F contains a locally free subsheaf F of rank d such that Proof From the previous statement we know that O n−d X ⊂ F. Take F = F/O n−d X . It is easy to see that F is also locally free sheaf of rank d. Moreover we have a short exact sequence This gives the following exact sequence Since k[X ] modules (X, O n−d X ), (X, F), (X, F ) are projective ( it is true locally hence also globally!) we have Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.