On the fixed locus of framed instanton sheaves on $\mathbb{P}^{3}$

TL;DR

This paper characterizes the fixed points of a torus action on the moduli space of rank 2 framed instanton sheaves on projective 3-space, showing they are non-locally free and relating them to stable pairs.

Contribution

It proves that fixed instanton sheaves are non-locally free with trivial double duals and classifies their support via stable pairs, providing bounds on the moduli space components.

Findings

Fixed locus consists only of non-locally free instanton sheaves.

Double duals of these sheaves are trivial bundles.

Provides a lower bound on the number of components in the fixed locus.

Abstract

Let $\mathbb{T}$ be the three dimensional torus acting on $\mathbb{P}^{3}$ and $\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c)$ be the fixed locus of the corresponding action on the moduli space of rank $2$ framed instanton sheaves on $\mathbb{P}^{3}.$ In this work, we prove that $\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c)$ consist only of non locally-free instanton sheaves whose double dual is the trivial bundle $\mathcal{O}_{\mathbb{P}^{3}}^{\oplus 2}$. Moreover, We relate these instantons to Pandharipande-Thomas stable pairs and give a classification of their support. This allows to compute a lower bound on the number of components of $\mathcal{M}^{\mathbb{T}}_{\mathbb{P}^{3}}(c).$