On the Bertini theorem in arbitrary characteristic

Abstract

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

1 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 $PGL(n)$ on the projective space $\mathbb P ^n$) 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 ${\mathcal{F}}$ is a locally free sheaf on $X$, with sufficiently many sections, then a generic section of $\mathcal{F}$ is transversal to $X.$ As a Corollary we give a proof of the Atiyah–Serre Splitting Theorem (compare with [1, 6] and [5]).

2 Notations and definitions

We assume for simplicity, that the base field $k$ is algebraically closed. Let $X$ be algebraic variety and $\mathcal{F}$ be a locally free sheaf on $X$ of rank $r.$ Take a section $s\in \Gamma (X,\mathcal{F}).$ 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 $\mathcal{F}= \mathcal{O}_U^r$ is trivial. Hence $s=(s_1,\ldots ,s_r)$ where $s_i\in \Gamma (U, O_U)=k[U].$ Then $s^{-1}(0)$ is given by the ideal $(s_1,\ldots ,s_r)\subset k[U].$ 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 $s^{-1}(0)=\emptyset .$

Let $X,Y$ be smooth varieties and $f:X\rightarrow Y$ be a morphism. We say that $f$ is unramified, if $f$ separates infinitely near points of $X$, i.e., the mapping $d_xf : T_x X\rightarrow T_{f(x)} Y$ is a monomorphism for every closed point $x\in X$ (see [3], p. 15).

If $X$ is an affine variety we will denote by $k[X]$ the ring $\Gamma (X,O_X)$. If $M$ is a $k[X]$ module then by $M^\sim$ we denote the sheafification of $M$—see [2], Definition on the page $110$.

3 Main result

Theorem 3.1

(Bertini theorem in arbitrary characteristic) Let $X$ be a smooth algebraic variety of dimension $d$ and let $\mathcal{F}$ be an invertible sheaf on $X$. Assume that $\mathcal{F}$ is generated by global sections $s_1,\ldots ,s_r\in \Gamma (X,\mathcal{F}).$ Assume that the morphism $\Phi$ from $X$ to the projective space $\mathbb P ^{r-1}$ given by $s_1,\ldots ,s_r$ is unramified. Then there is a Zariski open non-empty subset $U\subset k^r$ such that for every $c=(c_1,\ldots ,c_r)\in U$ the zero set of the section $s=\sum ^r_{i=1} c_i s_i$ is smooth.

Proof

Since $X$ is quasi-compact we can assume that $X$ is affine and the sheaf $\mathcal{F}$ is trivial. Hence we can identify $\mathcal{F}$ with $\mathcal{O}_X$ and now $s_i: X\rightarrow k$ are regular functions. Additionally we can assume that $s_r\equiv 1.$ Indeed, since $s_1,\ldots ,s_r$ generates $\mathcal{F}$ we have that open (affine!) subsets $U_i:=X{\setminus } \{s_i=0\}$ cover $X.$ Consequently we can assume that $s_r\not =0$ in $X.$ Take $s_i^{\prime }=s_i/s_r, \ i=1,\ldots ,r.$ If we prove our theorem for $s_i^{\prime }$, then we automatically prove it also for originals $s_i.$ Indeed, note that for a fixed $c=(c_1,\ldots ,c_r)\in k^r$ the ideals $(\sum ^r_{i=1} c_is_i)$ and $(\sum ^r_{i=1} c_is^{\prime }_i)$ are equal in $k[X]$. Moreover, we can assume that on $X$ we have global local coordinates $x_1,\ldots ,x_d$, i.e., the mapping $(x_1,\ldots ,x_d): X\rightarrow k^d$ is etale.

Let

$$\begin{aligned} V=\left\{ (c,x)\in k^r\times X : c_r+\sum ^{r-1}_{i=1} c_i s_i(x)=0\right\} . \end{aligned}$$

The variety $V$ is smooth. Indeed, let $(g_1,\ldots ,g_r)$ be the set of generators of the ideal $I(V)\subset k[X][c_1,\ldots ,c_r]$ and let $J(V)(c,x)=[\frac{\partial g_i}{\partial z_j}(c,x)],$ where $z=(c_1,\ldots ,c_r,x_1,\ldots ,x_d)$ is a set of global local coordinates on $k^r\times X.$ Let us note that a polynomial $h= \sum ^r_{i=1} c_i s_{i}(x)$ does belong to the ideal $I(V).$ Now we see that rank $J(V)\ge 1$, because partial derivatives of $h$ with respect to $c_i$ form a matrix $[s_{i}(x)]_{1\le i \le r},$ which has a rank $1$. Hence $\dim T_{(c,x)}V\le d+r-1.$ On the other hand $\dim V=d+r-1$ and consequently we have the equality $\dim T_{(c,x)}V = d+r-n=\dim V.$

Consider the projection:

$$\begin{aligned} q : V\in (c,x)\rightarrow c\in k^r. \end{aligned}$$

We show that for generic $c\in k^r$ the mapping $q$ is smooth on the set $q^{-1}(U_c)$, where $U_c$ is suitable neighborhood of $c.$

Indeed, let us compute the tangent space at $(c,x).$ It is given by the equation

$$\begin{aligned} dc_r+\sum ^{r-1}_{i=1} s_i(x)dc_i+ \sum _j\left(\,\sum _i^{r-1} c_i \frac{\partial s_i(x)}{\partial x_j}\right) dx_j. \end{aligned}$$

Let us note that rank $[\frac{\partial s_i(x)}{\partial x_j}]_{1\le i \le r-1, 1\le j\le d}=d.$ Indeed the mapping $\Phi : X\ni x \rightarrow (s_1(x),\ldots ,s_{r-1}(x))\in k^{r-1}$ is unramified.

Put $L_c:=\{ x\in X : \sum _{i=1}^{r-1} c_i \frac{\partial s_i(x)}{\partial x_j}=0, j=1,\ldots , d\}.$ Let us note that the mapping $q$ is not a submersion in a neighborhood of $q^{-1}(c)$ a exactly if $L_c\not =\emptyset .$ We show that the set $S:=\{c\in k^r: L_c\not =\emptyset \}$ is a constructible subset of $k^r$ of dimension less than $r.$

Indeed, let $W=\{ (c^{\prime },x)\in k^{r-1}\times X : \sum _{i=1}^{r-1} c_i \frac{\partial s_i(x)}{\partial x_j}=0, j=1,\ldots , d\}.$ Let $\pi : W\ni (c,x)\rightarrow x\in X.$ Let us note that the fiber of $\pi$ is a linear subspace of $k^{r-1}$ of dimension $r-1-d$ ( as a kernel of a suitable linear mapping). Hence $\dim W\le r-1.$

Let $W^{\prime }=\{ (c,x)\in V : \sum _{i=1}^{r-1} c_i \frac{\partial s_i(x)}{\partial x_j}=0, j=1,\ldots , d\}.$ We have a surjective mapping

$$\begin{aligned} s: W\ni (c^{\prime },x)\rightarrow \left(\left(c^{\prime }, -\sum ^{r-1}_{i=1} c_i^{\prime }s_i(x)\right),x\right)\in W^{\prime }. \end{aligned}$$

Since $S=\rho (W^{\prime })$, where $\rho : W^{\prime }\ni (c,x)\rightarrow c\in k^r$ and $\dim W=\dim s(W)=\dim W^{\prime }$ we have $\dim cl(S)\le r-1.$

This means that the projection $q: V\rightarrow k^r$ is a submersion outside a proper algebraic subset $\overline{S}\subset k^r.$ In particular the zero set of a generic section $s=\sum ^r_{i=1} c_i s_i$ is smooth. $\square$

Using a similar method we can generalize this result to higher dimension. As a Corollary we obtain Atiyah-Serre Spliting Theorem.

Theorem 3.2

Let $X$ be an algebraic variety of dimension $d$ and let $\mathcal{F}$ be a locally-free sheaf on $X$ of rank $n.$ Assume that $\mathcal{F}$ is generated by global sections $s_1,\ldots ,s_r\in \Gamma (X,\mathcal{F}).$ Then there is a Zariski open non-empty subset $U\subset k^r$ such that for every $c=(c_1,\ldots ,c_r)\in U$ the zero set of the section $s=\sum ^r_{i=1} c_i s_i$ is either empty or it has dimension $d -n.$ Moreover, if char $k=0$ and $X$ is smooth, then a generic section $s=\sum ^r_{i=1} c_i s_i$ is transversal to $X.$

Proof

Since $X$ is quasi-compact we can assume that $X$ is affine and the sheaf $\mathcal{F}$ is trivial, i.e., we can identify $\mathcal{F}$ with $\mathcal{O}_X^n$. Moreover we can assume that on $X$ we have a global local coordinates $(x_1,\ldots ,x_d).$ Hence $s_i=(s_{i1},\ldots ,s_{i,n}): X\rightarrow k^n$ are regular mappings. Let

$$\begin{aligned} V=\left\{ (c,x)\in k^r\times X : \sum ^r_{i=1} c_i s_i(x)=0\right\} . \end{aligned}$$

Let us note that $\dim V= r+d-n,$ where $d=\dim X.$ Indeed, we have a surjection $\pi : V\ni (c,x)\rightarrow x \in X.$ Any fiber of $\pi$ is a linear subspace of $k^r$ of dimension $r-n$ ( it is a kernel of surjective linear mapping $F_x: k^r\ni c\rightarrow \sum ^r_{i=1} c_is_i(x)\in \mathbf{E}^n_x,$ where $\mathbf{E}^n=X\times k^n$ is a trivial vector bundle of rank $n$). Consequently $\dim V=\dim X+ r-n=d+r-n.$ Now consider the second projection:

$$\begin{aligned} q : V\in (c,x)\rightarrow c\in k^r. \end{aligned}$$

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 $(g_1,\ldots ,g_r)$ be the set of generators of the ideal $I(V)\subset k[X][c_1,\ldots ,c_r]$ and let $J(V)(c,x)=[\frac{\partial g_i}{\partial z_j}(c,x)],$ where $z=(c_1,\ldots ,c_r, x_1,\ldots , x_d)$ is a set of global local coordinates on $k^r\times X.$ Let us note that polynomials $h_j= \sum ^r_{i=1} c_i s_{ij}(x)$ does belong to the ideal $I(V).$ Now we see that rank $J(V)\ge n$, because partial derivatives of $h_j$ with respect to $c_i$ form a matrix $[s_{ij}(x)]_{1\le i \le r, 1\le j\le n},$ which has a rank $n$. Hence $\dim T_{(c,x)}V\le d+r-n.$ 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). $\square$

Corollary 3.3

(Atiyah–Serre Splitting Theorem) Let $X$ be an algebraic variety of dimension $d$ and let $\mathcal{F}$ be a locally free sheaf on $X$ of rank $n.$ Assume that $\mathcal{F}$ is generated by global sections $s_1,\ldots ,s_r\in \Gamma (X,\mathcal{F}).$ If $n>d$ then $\mathcal{F}$ contains a trivial subsheaf $\mathcal A\subset \mathcal{F}$ of rank $n-d.$

Proof

Indeed, by induction we can find $n-d$ linearly independent sections $t_j=\sum ^r_{i=1} c_{j,i} s_i$ (note that the quotient sheaf of $\mathcal{F}$ is generated by the same sections (or rather their classes) as $\mathcal{F}$). $\square$

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 $\mathcal{F}$ be a locally free sheaf on $X$ of rank $n.$ If $n>d$ then $\mathcal{F}$ contains a locally free subsheaf $\mathcal{F}^{\prime }$ of rank $d$ such that

$$\begin{aligned} {\mathcal{F}}={\mathcal{F}}^{\prime } \oplus {\mathcal{O}}_X^{n-d}. \end{aligned}$$

Proof

From the previous statement we know that $\mathcal{O}_X^{n-d}\subset \mathcal{F}.$ Take $\mathcal{F}^{\prime }=\mathcal{F}/\mathcal{O}_X^{n-d}.$ It is easy to see that $\mathcal{F}^{\prime }$ is also locally free sheaf of rank $d.$ Moreover we have a short exact sequence

$$\begin{aligned} 0\rightarrow \mathcal{O}_X^{n-d}\rightarrow \mathcal{F}\rightarrow \mathcal{F}^{\prime }\rightarrow 0. \end{aligned}$$

This gives the following exact sequence

$$\begin{aligned} 0\rightarrow \Gamma (X,\mathcal{O}_X^{n-d})\rightarrow \Gamma (X,\mathcal{F})\rightarrow \Gamma (X, \mathcal{F}^{\prime })\rightarrow H^1(X,\mathcal{O}_X^{n-d})=0. \end{aligned}$$

Since $k[X]$ modules $\Gamma (X,\mathcal{O}_X^{n-d}), \Gamma (X,\mathcal{F}), \Gamma (X, \mathcal{F}^{\prime })$ are projective ( it is true locally hence also globally!) we have

$$\begin{aligned} \Gamma (X,\mathcal{F})= \Gamma (X, \mathcal{F}^{\prime })\oplus \Gamma (X,\mathcal{O}_X^{n-d}). \end{aligned}$$

Finally

$$\begin{aligned} \Gamma (X,\mathcal{F})^{\sim } = \Gamma (X, \mathcal{F}^{\prime })^{\sim }\oplus \Gamma (X,\mathcal{O}_X^{n-d})^{\sim }, \end{aligned}$$

and we have

$$\begin{aligned} \mathcal{F}=\mathcal{F}^{\prime } \oplus \mathcal{O}_X^{n-d} \end{aligned}$$

(see [2], Prop. 5.1, p. 110). $\square$