Universal plane curve and moduli spaces of 1-dimensional coherent sheaves

TL;DR

This paper demonstrates that the universal plane curve of degree d > 2 can be embedded into a Simpson moduli space of 1-dimensional sheaves, revealing geometric structures and singularities related to sheaves not locally free on their support.

Contribution

It establishes a new connection between universal plane curves and Simpson moduli spaces, providing a geometric interpretation of singular loci and a natural compactification via blow-up.

Findings

Universal plane curve is a closed subvariety of a Simpson moduli space.

Singular locus corresponds to sheaves not locally free on their support.

Blow-up along the singular locus yields a natural compactification.

Abstract

We show that the universal plane curve M of fixed degree d > 2 can be seen as a closed subvariety in a certain Simpson moduli space of 1-dimensional sheaves on a projective plane contained in the stable locus. The universal singular locus coincides with the subvariety of M consisting of sheaves that are not locally free on their support. It turns out that the blow up of M along M' may be naturally seen as a compactification of M_B = M\M' by vector bundles (on support).