On the singular sheaves in the fine Simpson moduli spaces of $1$-dimensional sheaves

TL;DR

This paper investigates the structure of certain singular sheaves within the Simpson moduli space of semi-stable sheaves on a projective plane, revealing their codimension and geometric modifications for degrees four and above.

Contribution

It characterizes the singular subvariety of non-locally free sheaves in the moduli space and interprets the blow-up as a vector bundle modification.

Findings

The subvariety of non-locally free sheaves has codimension 2 for d ≥ 4.

The blow-up along this subvariety provides a geometric modification of the moduli space.

The study enhances understanding of the singularities in Simpson moduli spaces.

Abstract

In the Simpson moduli space $M$ of semi-stable sheaves with Hilbert polynomial $dm-1$ on a projective plane we study the closed subvariety $M'$ of sheaves that are not locally free on their support. We show that for $d\ge 4$ it is a singular subvariety of codimension $2$ in $M$. The blow up of $M$ along $M'$ is interpreted as a (partial) modification of $M\setminus M'$ by vector bundles (on support).