A Mehta-Ramanathan theorem for linear systems with basepoints

TL;DR

This paper extends the Mehta-Ramanathan theorem to include restrictions of semistable sheaves to curves passing through specified points, under certain smoothness and local freeness conditions, with applications to semipositivity.

Contribution

It generalizes the Mehta-Ramanathan theorem to sheaves with basepoints, providing new conditions for the preservation of semistability upon restriction.

Findings

Semistability is preserved on general complete intersection curves passing through finite sets of points.

The result applies to sheaves with locally free Jordan-Hölder factors at specified points.

A generalization of Miyaoka's semipositivity theorem is obtained.

Abstract

Let $(X, H)$ be a normal complex projective polarized variety and $\mathscr E$ an $H$-semistable sheaf on $X$. We prove that the restriction $\mathscr E\big|_C$ 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-H\"older filtration of $\mathscr E$ are locally free. As an application, we obtain a generalization of Miyaoka's generic semipositivity theorem.