Let $X$ be a smooth projective variety of dimension $\geq 2$ and $E$ a vector bundle on $X$ of rank $r\geq 2$. Is it true that, if $E$ is globally generated, then the zero locus of a general section of $E$ has codimension $\geq r$ in $X$?

$\begingroup$ Quick correction: You want $r$ to be the rank of $E$, not the dimension of $X$. $\endgroup$ – David E Speyer May 4 '11 at 12:22

$\begingroup$ sorry! Of course you are right! $\endgroup$ – ginevra86 May 4 '11 at 12:31

6$\begingroup$ If $E$ is trivial, this isn't true because the zero sets are empty for general sections. You need to make $E$ "sufficiently positive". Precise criteria can probably found in Lazarsfeld's book on positivity. $\endgroup$ – Donu Arapura May 4 '11 at 12:40

$\begingroup$ Actually I am interested in knowing whether the codimension of the zero locus is at least $r$. I was imprecise when formulating the question. $\endgroup$ – ginevra86 May 4 '11 at 12:56

$\begingroup$ I think this is Lemma 5.2 in "3264 and All That". $\endgroup$ – Hoot May 9 '16 at 23:27
As Donu pointed out, it may happen that the general section has no zeroes. Anyway, the the answer to your question is yes.
This is a special case of the following more general result "of Bertini type" about degeneracy loci of morphism of vector bundles, whose proof can be found in Ottaviani's book Varietà proiettive di codimensione piccola. This book is in italian; if you prefer a treatment in english, look at Lazarsfeld's book Positivity in algebraic geometry II, Section 7.2, and at the references given therein.
Theorem. Let $\phi \colon F \to E$ be a morphism between vector bundles on $X$, with $\textrm{rank} \, F=s$, $\textrm{rank} \, E=r$, and let $D_k(\phi)$ be the set of points $x \in X$ such that $\phi_x \colon F_x \to E_x$ has rank $\leq k$.
If $F^{*} \otimes E$ is generated by global sections, then for the general $\phi$ either $D_k(\phi)$ is empty or it has the expected codimension $(sk)(rk)$.
Moreover, if $F^{*} \otimes E$ is ample and $(sk)(rk) < \dim X$, then for the general $\phi$ the locus $D_k(\phi)$ is nonempty, of the expected codimension $(sk)(rk)$.
Now apply this result to the morphism $$\phi \colon \mathcal{O}_X \to E$$ induced by the section. Since $D_0(\phi)$ is exactly the locus where the section vanishes, you are done.

$\begingroup$ If a section has no zeros then the codimension of its zero locus is in fact $\ge r$... $\endgroup$ – diverietti May 4 '11 at 21:44

$\begingroup$ You are right, but the question was edited. Ok, I can edit the answer too... $\endgroup$ – Francesco Polizzi May 5 '11 at 6:40

$\begingroup$ Could you give an English reference for this result? I have hard time finding this book. $\endgroup$ – xyzzyz Feb 23 '14 at 17:43