Let $E$ be a vector bundle on a smooth projective variety $X$ and assume $\mathrm{rk}(E)>\mathrm{dim}(X)$. If $E$ is globally generated, then a general section of $E$ is nowhere vanishing (see Question [Vanishing locus of a general section of a vector bundle.](https://mathoverflow.net/questions/63909/vanishing-locus-of-a-general-section-of-a-vector-bundle)). Does the same hold true under the milder hypothesis that $E$ is (only) generically generated by global sections?



Generically generated vector bundles & degeneracy loci.
It seems to me that the answer is *no*, because of the following counterexample.


Take a surface $X$ with a linear pencil $|L|$ having a unique base point $x$. Now set $$E=\mathcal{O}\_X(L) \oplus \mathcal{O}\_X(L) \oplus \mathcal{O}\_X(L).$$
Since $L$ is generically globally generated, the same is true for $E$.


On the other hand, any section of $E$ is given by a vector of the form $s=(s\_1, s\_2, s\_3)$ where each $s\_i$ is a section of $L$. By assumption we have $s\_i(x)=0$ for $i\in \{1,2,3\}$, so $s(x)=0$. In other words, all sections of $E$ vanish at the point $x$.


**Example.** Let $p\_1, p\_2, p\_3, p$ be the four base points of a pencil of conics in $\mathbb{P}^2$ and let $\pi \colon X \to \mathbb{P}^2$ be the blow-up of $\mathbb{P}^2$ at $p\_1, p\_2, p\_3$. Set $L=\pi^\*\mathcal{O}\_{P^2}(2)-E\_1-E\_2-E\_3$, where the $E\_i$ are the exceptional divisors. Then $|L|$ is a pencil in $X$ with a unique base point, namely the point $x=\pi^{-1}(p)$.


