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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 $X$?

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  • $\begingroup$ Quick correction: You want $r$ to be the rank of $E$, not the dimension of $X$. $\endgroup$ May 4, 2011 at 12:22
  • $\begingroup$ sorry! Of course you are right! $\endgroup$
    – ginevra86
    May 4, 2011 at 12:31
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    $\begingroup$ 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. $\endgroup$ May 4, 2011 at 12:40
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    $\begingroup$ 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, 2011 at 12:56
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    $\begingroup$ I think this is Lemma 5.2 in "3264 and All That". $\endgroup$
    – Hoot
    May 9, 2016 at 23:27

1 Answer 1

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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 $(s-k)(r-k)$.

Moreover, if $F^{*} \otimes E$ is ample and $(s-k)(r-k) < \dim X$, then for the general $\phi$ the locus $D_k(\phi)$ is non-empty, of the expected codimension $(s-k)(r-k)$.

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.

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  • $\begingroup$ If a section has no zeros then the codimension of its zero locus is in fact $\ge r$... $\endgroup$
    – diverietti
    May 4, 2011 at 21:44
  • $\begingroup$ You are right, but the question was edited. Ok, I can edit the answer too... $\endgroup$ May 5, 2011 at 6:40
  • $\begingroup$ Could you give an English reference for this result? I have hard time finding this book. $\endgroup$
    – xyzzyz
    Feb 23, 2014 at 17:43

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