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

Vincenzo Antonelli, Francesco Malaspina, Simone Marchesi, Joan Pons-Llopis

TL;DR

The paper investigates instanton bundles on the flag threefold $F=F(0,1,2)$, introducing $D$-'t Hooft and special $t$-Hooft bundles and establishing the existence of μ-stable examples for all charges $k$. It develops a comprehensive geometric framework: a detailed study of the flag geometry, the Hilbert scheme of degree-six del Pezzo surfaces in $F$, and the role of rational and elliptic curves as zero loci of sections of instantons via Serre correspondence and monads. The authors classify moduli spaces of these bundles, showing intricate stratifications, multiple components, and singularities tied to special loci, and they prove precise splitting behavior of instanton bundles on conics. These results advance the understanding of instanton geometry on twistor-like flag varieties and provide explicit, computable descriptions of their moduli and associated curve data.

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 $μ$-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.

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

TL;DR

The paper investigates instanton bundles on the flag threefold , introducing -'t Hooft and special -Hooft bundles and establishing the existence of μ-stable examples for all charges . It develops a comprehensive geometric framework: a detailed study of the flag geometry, the Hilbert scheme of degree-six del Pezzo surfaces in , and the role of rational and elliptic curves as zero loci of sections of instantons via Serre correspondence and monads. The authors classify moduli spaces of these bundles, showing intricate stratifications, multiple components, and singularities tied to special loci, and they prove precise splitting behavior of instanton bundles on conics. These results advance the understanding of instanton geometry on twistor-like flag varieties and provide explicit, computable descriptions of their moduli and associated curve data.

Abstract

In this work we study the moduli spaces of instanton bundles on the flag twistor space . We stratify them in terms of the minimal twist supporting global sections and we introduce the notion of (special) 't Hooft bundle on . In particular we prove that there exist -stable 't Hooft bundles for each admissible charge . We completely describe the geometric structure of the moduli space of (special) 't Hooft bundles for arbitrary charge . Along the way to reach these goals, we describe the possible structures of multiple curves supported on some rational curves in as well as the family of del Pezzo surfaces realized as hyperplane sections of . Finally we investigate the splitting behaviour of 't Hooft bundles when restricted to conics.
Paper Structure (13 sections, 41 theorems, 171 equations)

This paper contains 13 sections, 41 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. The geometry of the flag variety
  3. Description of the Hilbert scheme of degree six del Pezzo surfaces in $F$
  4. A local description
  5. A global description
  6. Multiple rational curves on the Flag variety
  7. Quasi-primitive extensions
  8. Multiple structures on lines
  9. $h_i$-'t Hooft instantons
  10. $h$-'t Hooft bundles
  11. Special 't Hooft bundles
  12. Moduli spaces of 't Hooft bundles
  13. The splitting type of an instanton bundle on conics

Key Result

Theorem A

For each $k\geq 1$ there exist $\mu$-stable $h_i$-'t Hooft bundles and special instanton bundles of charge $k$. Moreover, for each $k\geq 2$ there exist $\mu$-stable, proper $(h_1+h_2)$-'t Hooft bundles.

Theorems & Definitions (89)

  • Theorem A
  • Theorem B
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • ...and 79 more