Instanton sheaves on Fano threefolds

TL;DR

This paper extends the concept of instanton sheaves to Fano threefolds, classifies rank 1 instanton sheaves, and analyzes the moduli space of rank 2 instanton sheaves, revealing their structure and connectedness.

Contribution

It generalizes instanton sheaves to arbitrary rank on Fano threefolds, classifies rank 1 cases, and describes the moduli space of rank 2 instanton sheaves on a quadric threefold.

Findings

Rank 1 instanton sheaves are classified.

All curves with rank 0 instanton sheaves are described.

The moduli space of rank 2 instanton sheaves of charge 2 is connected.

Abstract

Generalizing the definitions originally presented by Kuznetsov and Faenzi, we study (possibly non locally free) instanton sheaves of arbitrary rank on Fano threefolds. We classify rank 1 instanton sheaves and describe all curves whose structure sheaves are rank 0 instanton sheaves. In addition, we show that every rank 2 instanton sheaf is an elementary transformation of a locally free instanton sheaf along a rank 0 instanton sheaf. To complete the paper, we describe the moduli space of rank 2 instanton sheaves of charge 2 on a quadric threefold $X$, and show that the full moduli space of rank 2 semistable sheaves on $X$ with Chern classes $(c_1,c_2,c_3)=(-1,2,0)$ is connected and contains, besides the instanton component, just one other irreducible component which is also fully described.