A Mehta-Ramanathan theorem for linear systems with basepoints

Abstract

Let $(X,H)$ be a normal complex projective polarized variety and $\mathcal{E}$ an $H$-semistable sheaf on $X$. We prove that the restriction $\mathcal{E}{|}_C$ to a sufficiently positive general complete intersection curve $C\subset X$ passing through a prescribed finite set of points $S\subset X$ remains semistable, provided that at each $p\in S$, the variety $X$ is smooth and the factors of a Jordan-Hölder filtration of $\mathcal{E}$ are locally free. As an application, we obtain a generalization of Miyaoka's generic semipositivity theorem.

Mostly minor changes. Improved Example 1.9. To appear in Mathematische Nachrichten

References in corpus (2)

• The diminished base locus is not always closed
• On positivity and semistability of vector bundles in finite and mixed characteristics