# Semistable rank 2 sheaves with singularities of mixed dimension on $\mathbb{P}^3$

## TL;DR

This paper introduces new irreducible components of the moduli space of semistable rank 2 sheaves on projective 3-space, featuring sheaves with singularities of both zero and one-dimensional components, constructed via elementary transformations.

## Contribution

It provides the first examples of irreducible components with general sheaves having mixed-dimensional singularities in the Gieseker-Maruyama moduli scheme.

## Key findings

- New irreducible components of the moduli space identified.
- Sheaves with mixed-dimensional singularities constructed explicitly.
- First examples of such components with these properties.

## Abstract

We describe new irreducible components of the Gieseker-Maruyama moduli scheme $\mathcal{M}(3)$ of semistable rank 2 coherent sheaves with Chern classes $c_1=0,\ c_2=3,\ c_3=0$ on $\mathbb{P}^3$, general points of which correspond to sheaves whose singular loci contain components of dimensions both 0 and 1. These sheaves are produced by elementary transformations of stable reflexive rank 2 sheaves with $c_1=0,\ c_2=2,\ c_3=2$ or 4 along a disjoint union of a projective line and a collection of points in $\mathbb{P}^3$. The constructed families of sheaves provide first examples of irreducible components of the Gieseker-Maruyama moduli scheme such that their general sheaves have singularities of mixed dimension.