# The spectrum of torsion free sheaves on $\mathbb{P}^3$ and applications

## TL;DR

This paper investigates the spectrum of rank 2 torsion free sheaves on projective 3-space, providing classifications and examples of moduli space components with identical spectra, addressing a question by Rao.

## Contribution

It offers a complete description of the spectrum for specific Chern classes of semistable rank 2 torsion free sheaves on 3, revealing new irreducible components in the moduli space.

## Key findings

- Classified spectra for sheaves with Chern classes (-1,2,0) and (0,3,0)
- Constructed examples of distinct moduli components with same spectrum
- Answered Rao's question in the context of torsion free sheaves

## Abstract

We study the spectrum of rank $2$ torsion free sheaves on $\mathbb{P}^3$ with aim of producing examples of distinct irreducible components of the moduli space with the same spetrcum answering the question presented by Rao for the case of torsion free sheaves. In order to do so, we provide a full description of the spectrum of the sheaves in the moduli space of semistable rank $2$ torsion free sheaves on $\mathbb{P}^3$ with Chern classes $(c_1, c_2,c_3)$ equals to $(-1,2,0)$ and $(0,3,0)$.