Torsion free instanton sheaves on the blow-up of $\mathbb{P}^{3}$ at a point

TL;DR

This paper extends the concept of instanton bundles to include certain non-locally free sheaves on the blow-up of projective 3-space, proving key properties and constructing examples relevant to moduli space compactification.

Contribution

It introduces a new definition of instanton sheaves on the blow-up of -space, showing reflexive sheaves are locally free and analyzing torsion free sheaves with singularities, plus constructing examples and studying stability.

Findings

Reflexive instanton sheaves are necessarily locally free.

Strictly torsion free instanton sheaves have 1-dimensional singularities.

Constructed examples are smooth and smoothable.

Abstract

We consider an extension of the instanton bundles definition, given by Casnati-Coskun-Genk-Malaspina, for Fano threefolds, in order to include non locally-free ones on the blow-up $\widetilde{\mathbb{P}^{3}},$ of the projective $3-$space at a point. With the proposed definition, we prove that any reflexive instanton sheaf must be locally free, and that the strictly torsion free instanton sheaves have singularities of pure dimension $1.$ We construct examples and study their $\mu-$stability. Furthermore, these sheaves will play a role in (partially) compactifying the t'Hooft component of the moduli space of instantons, on $\widetilde{\mathbb{P}^{3}}.$ Finally, examples of these are shown to be smooth and smoothable.