On the moduli spaces of semi-stable plane sheaves of dimension one and multiplicity five

TL;DR

This paper characterizes the moduli spaces of semi-stable sheaves of dimension one and multiplicity five on the complex projective plane, providing resolutions, geometric descriptions, and stratifications.

Contribution

It introduces explicit locally free resolutions, geometric descriptions via extensions, and stratifications of the moduli spaces, with concrete quotient descriptions.

Findings

Resolutions of length one for all such sheaves

Geometric descriptions of sheaves via extensions

Stratifications of moduli spaces as quotient spaces

Abstract

We find locally free resolutions of length one for all semi-stable sheaves supported on curves of multiplicity five in the complex projective plane. In some cases we also find geometric descriptions of these sheaves by means of extensions. We give natural stratifications for their moduli spaces and we describe the strata as certain quotients modulo linear algebraic groups. In most cases we give concrete descriptions of these quotients as fibre bundles.