# Irreducible components of the moduli space of rank 2 sheaves of odd determinant on the projective space

## TL;DR

This paper identifies new irreducible components in the moduli space of rank 2 semistable sheaves on projective space, showing their growth, rationality, and connectedness, with explicit counts for certain Chern classes.

## Contribution

It describes new irreducible components of the moduli space, computes their number for specific Chern classes, and proves their rationality and connectedness.

## Key findings

- Number of components grows with second Chern class
- All components are rational
- Moduli spaces are connected and some sheaves are smoothable

## Abstract

We describe new irreducible components of the moduli space of rank $2$ semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes $(c_1,c_2,c_3)=(-1,2n,0)$ and $(c_1,c_2,c_3)=(0,n,0)$ always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, and compute the exact number of irreducible components of the moduli spaces of rank 2 semistable torsion free sheaves with Chern classes $(c_1,c_2,c_3)=(-1,2,m)$ for all possible values for $m$; all components turn out to be rational. Furthermore, we also prove that these moduli spaces are connected, showing that some of sheaves here considered are smoothable.