Hartshorne II Exercise 8.2

Question

Let $X$ be a projective variety over an algebraically closed field $k$, $\mathscr E$ be a locally free sheaf of rank $r$ on $X$, and $V\subset \mathscr E(X)$ be a subspace of its global sections which can generate $\mathscr E$. Show that there is an element $s\in V$, and that for each $x\in X$, we have $s_x\notin m_x\mathscr E_x$.

The hint is to go like the proof of the Bertini's theorem. Now I consider $V$ as a projective space. Consider a set $Z:=\{(x,v)|x\in X,v\in V,s.t. v_x\in m_x\mathscr E_x\}\subset X\times V$. I want to prove $Z$ is a closed subset(then we can just copy almost everything of the proof of Bertini's theorem). But I have no idea how to prove this detail... It seems not so direct to see it's closed, since we don't have any concrete description of elements of $V$. Could you offer some help?(And if I go the wrong way could you point it out?) Thanks!

Answer 1

$\def\cE{\mathcal{E}}\def\cO{\mathcal{O}}\def\AA{\mathbb{A}}\def\VV{\mathbb{V}}\def\rank{\operatorname{rank}}\def\Spec{\operatorname{Spec}}$

We first reduce to the case where $V$ is finite-dimensional. Let $r=\rank\cE$ and let $\eta\in X$ be the generic point. There are some finite number of elements $\{v_i\}_{i\in I}\subset V$ whose image in $\cE_\eta$ generate $\cE_\eta$ as a $\cO_{X,\eta}$-module. Let $\cE'$ be the quotient of $\cE$ by the image of the morphism $\cO_X^{\oplus I}\to \cE$ given by sending $e_i\mapsto v_i$. This is a quasi-coherent sheaf which is zero at the generic point by construction, so it's supported on some closed lower-dimensional set $X'$, which is a finite union of irreducible components by $X$ being noetherian. Now do the same thing for all the generic points of the irreducible components of $X'$: we add a finite list of elements of our vector space $V$ which generate the stalk of $\cE$ at each such generic point to form a collection $\{v_i\}_{i\in I'}$ and let $\cE''$ be the quotient of $\cE$ by the image of the morphism $\cO_X^{\oplus I'}\to \cE$ given by sending $e_i\mapsto v_i$. Eventually, by induction, the process terminates with a finite number of elements in our collection $\{v_i\}$ whose images generate $\cE_x$ for all $x\in X$. Therefore we may take $V$ finite dimensional.

Now consider the map of locally free sheaves $V\otimes\cO_X\to \cE$ given by sending $v\mapsto v$. This induces a map of geometric vector bundles $X\times_k V\to \VV(\cE)$ per the construction in exercise II.5.18. On an open affine subset $\Spec A\subset X$ where $\cE$ is trivial, the restriction of this map is isomorphic to an $A$-linear morphism $\AA^{\dim V}_A\to \AA^r_A$ given in coordinates by some $r\times(\dim V)$ matrix $M=\{m_{ij}\}$ with entries in $A$. Taking coordinates $x_1,\cdots,x_{\dim V}$ on $\AA^{\dim V}_A$, the locus $\{(x,s)\mid s|_x=0\}$ is then cut out inside $\AA^{\dim V}_A$ by the $r$ linear equations corresponding to the $r$ entries of the vector $M(x_1,\cdots,x_{\dim V})$, and this description is compatible with overlaps by the fact that the transition functions on $\VV(\cE)$ are linear. So $Y=\{(x,s)\mid s|_x=0\}\subset X\times_k V$ is a closed subscheme, and we see that its fiber over any closed point $x\in X$ is of dimension $\dim V-\rank \cE$ by the rank nullity theorem and the fact that we chose $V$ to generate $\cE$. Now you can apply the rest of the argument from Hartshorne's proof of Bertini.