Singular locus of instanton sheaves on $\mathbb{P}^3$

TL;DR

This paper investigates the structure of singular loci in rank 2 instanton sheaves on projective 3-space, establishing their pure dimension and properties of their duals, with examples showing these results do not extend to higher ranks.

Contribution

It proves that the singular locus of non-locally free rank 2 instanton sheaves on P^3 has pure dimension 1 and describes properties of their duals and Ext sheaves, highlighting differences in higher ranks.

Findings

Singular locus of rank 2 instanton sheaves has pure dimension 1.

Dual and double dual sheaves are locally free instantons.

Counterexamples exist for higher rank instanton sheaves.

Abstract

We prove that the singular locus of a rank 2 instanton sheaf $E$ on $\mathbb{P}^3$ which is not locally free has pure dimension 1. Moreover, we also show that the dual and double dual of $E$ are isomorphic locally free instanton sheaves, and that the sheaves $\mathcal{E}xt^1(E,\mathcal{O}_{\mathbb{P}^3)$ and $E^{\vee\vee}/E$ are rank $0$ instantons. We also provide explicit examples of instanton sheaves of rank $3$ and $4$ illustrating that all of these claims are false for higher rank instanton sheaves.