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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.). Does the same hold true under the milder hypothesis that $E$ is (only) generically generated by global sections?

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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 $x=\pi^{-1}(p)$.

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