On the geometry of the moduli spaces of semistable sheaves supported on plane quartics

TL;DR

This paper analyzes the structure of moduli spaces of semistable sheaves supported on plane quartics, decomposing them into subvarieties and describing their resolutions to understand their geometric properties.

Contribution

It provides a detailed decomposition of moduli spaces of semistable sheaves on the plane and characterizes each component via resolutions and quotient descriptions.

Findings

Decomposition of moduli spaces into locally closed subvarieties.

Explicit descriptions of sheaves via locally free resolutions.

Identification of quotient structures for each subvariety.

Abstract

We decompose each moduli space of semistable sheaves on the complex projective plane with support of dimension one and degree four into locally closed subvarieties, each subvariety being the good or geometric quotient of a set of morphisms of locally free sheaves modulo a reductive or a nonreductive group. We find locally free resolutions of length one of all these sheaves and describe them.