Framed instanton pairs on the blow-up of the projective 3-space at a point

TL;DR

The paper addresses the construction and stability of framed sheaves in the setting of the blow-up $\widetilde{\mathbb{P}^{3}}$ by framing along a divisor to a fixed instanton $\mathcal{E}_{D}$. It extends the Huybrechts–Lehn framework to threefolds, proving $\mu$-stability for framed instantons and the existence of a fine, quasi-projective moduli space for the associated moduli functor. It then specializes to $t'Hooft$ instantons on $\widetilde{\mathbb{P}^{3}}$, showing that the moduli locus of locally free framed $t'Hooft$ pairs is open and unobstructed, with obstruction vanishing given by $Ext^2(\mathcal{E},\mathcal{E}(-1,0))=0$. Overall, the work provides a concrete higher-dimensional example of framed instanton moduli and connects Hartshorne-Serre constructions to moduli theory on a blow-up threefold. This advances the understanding of framed sheaves beyond curves and surfaces and offers a new geometric setting for higher-dimensional gauge-theoretic objects.

Abstract

We study some Huybrechts and Lehn framed sheaves on the Fano 3-fold given by blowing-up the 3-projective space at a point. In contrast with the cases of curves and surfaces, there are very few examples in higher dimensions. In this notes we give a new example of such pairs in dimension 3 and prove that the moduli space under study is fine, quasi-projective and unobstructed.

Theorems & Definitions (27)

• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.6
• proof
• Lemma 2.7
• proof
• Lemma 2.8
• proof