Minimal bundles and fine moduli spaces

TL;DR

This paper investigates minimal sheaves on smooth projective curves, showing they define ample divisors on the Picard torus, classifying low-rank cases, and relating moduli spaces to Quot schemes.

Contribution

It introduces the concept of minimal sheaves, classifies them for ranks one and two, and connects moduli spaces of rank two bundles to Quot schemes.

Findings

Minimal sheaves define ample divisors on Picard torus.

Complete classification of minimal sheaves of rank one and two.

Moduli space of rank two bundles of odd degree is a Quot scheme.

Abstract

We study sheaves E on a smooth projective curve X which are minimal with respect to the property that $h^0(E \otimes L) >0$ for all line bundles L of degree zero. We show that these sheaves define ample divisors D(E) on the Picard torus Pic(X). Next we classify all minimal sheaves of rank one and two. As an application we show that the moduli space parameterizing rank two bundles of odd degree can be obtained as a Quot scheme.