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?


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.