't Hooft bundles on the complete flag threefold and moduli spaces of instantons

TL;DR

This paper investigates the structure of moduli spaces of instanton bundles on the flag threefold, introducing 't Hooft bundles, and describes their geometric properties, stability, and behavior under restrictions.

Contribution

It introduces the notion of 't Hooft bundles on the flag threefold and fully describes their moduli spaces for all charges, including stability and geometric structure.

Findings

Existence of μ-stable 't Hooft bundles for each admissible charge

Complete description of the moduli space of 't Hooft bundles

Analysis of splitting behavior on conics

Abstract

In this work we study the moduli spaces of instanton bundles on the flag twistor space $F:=F(0,1,2)$. We stratify them in terms of the minimal twist supporting global sections and we introduce the notion of (special) 't Hooft bundle on $F$. In particular we prove that there exist $\mu$-stable 't Hooft bundles for each admissible charge $k$. We completely describe the geometric structure of the moduli space of (special) 't Hooft bundles for arbitrary charge $k$. Along the way to reach these goals, we describe the possible structures of multiple curves supported on some rational curves in $F$ as well as the family of del Pezzo surfaces realized as hyperplane sections of $F$. Finally we investigate the splitting behaviour of 't Hooft bundles when restricted to conics.