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Ulrich and Instanton Bundles on Special Cubic Fourfolds

Gianfranco Casnati, Daniele Faenzi, Federica Galluzzi

Abstract

We study instanton and Ulrich bundles on hypersurfaces of the projective space, with a focus on special cubic fourfolds and generalized Pfaffians, notably defined by skew-symmetric endomorphisms of Steiner bundles. We prove that the acyclic extensions of instantons deform to Ulrich bundles and deduce that the existence of instantons of low rank and charge implies the existence of Ulrich bundles of low rank, which in turn forces the fourfold to lie in some Hassett divisor. Finally we take a closer look to divisors of cubics with discriminant 18 and 20.

Ulrich and Instanton Bundles on Special Cubic Fourfolds

Abstract

We study instanton and Ulrich bundles on hypersurfaces of the projective space, with a focus on special cubic fourfolds and generalized Pfaffians, notably defined by skew-symmetric endomorphisms of Steiner bundles. We prove that the acyclic extensions of instantons deform to Ulrich bundles and deduce that the existence of instantons of low rank and charge implies the existence of Ulrich bundles of low rank, which in turn forces the fourfold to lie in some Hassett divisor. Finally we take a closer look to divisors of cubics with discriminant 18 and 20.

Paper Structure

This paper contains 29 sections, 33 theorems, 149 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Notation and conventions
  4. Reminder on Hartshorne-Serre's construction
  5. Instanton sheaves
  6. Pfaffians representations and rank-2 instantons on hypersurfaces
  7. Instanton bundles on hypersurfaces and associated surfaces
  8. Pfaffians and discriminants
  9. Instantons and Ulrich complexity of special cubic fourfolds
  10. Ulrich bundles and the Kuznetsov component
  11. Ulrich bundles and instantons
  12. Unobstructed instantons
  13. Rank two instanton bundles on smooth cubic hypersurfaces
  14. General cubic hypersurfaces in ${\mathcal{C}}_{14}$.
  15. Cubic hypersurfaces in ${\mathcal{C}}_{18}$.
  16. ...and 14 more sections

Key Result

theorem 1.1

Let ${\mathcal{E}}$ be an unobstructed instanton sheaf on a smooth cubic fourfold $X$. Then the moduli space of simple sheaves on $X$ is smooth at the point $[{\mathcal{E}}]$.

Theorems & Definitions (78)

  • theorem 1.1
  • theorem 1.2
  • theorem 1.2
  • theorem 1.3
  • theorem 1.4
  • theorem 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 68 more