Raynaud vector bundles

TL;DR

This paper constructs special vector bundles on smooth projective curves that characterize semistability of other sheaves via homomorphisms, and provides bounds on linear systems with base points on moduli spaces.

Contribution

It introduces a new construction of vector bundles that detect semistability and offers effective bounds on linear systems on moduli spaces.

Findings

Vector bundles $R^r_$ characterize semistability via homomorphisms.

Effective bounds on $r$ for base points in linear systems on moduli spaces.

New tools for studying stability and linear systems on algebraic curves.

Abstract

We construct vector bundles $R^r_\mu$ on a smooth projective curve $X$ having the property that for all sheaves $E$ of slope $\mu$ and rank $r$ on $X$ we have an equivalence: $E$ is a semistable vector bundle $\iff$ $Hom(R^r_\mu,E)=0$. As a byproduct of our construction we obtain effective bounds on $r$ such that the linear system $|R \cdot \Theta|$ has base points on the moduli space $U_X(r,r(g-1))$.