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 $X$? 



Vanishing locus of a general section of a vector bundle.
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.



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> **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 $\leq k$.
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> 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 $(s-k)(r-k)$.
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> 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 $(s-k)(r-k)$.
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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.


