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Cubic fourfolds of discriminant 24 and rationality

Brendan Hassett

Abstract

Cubic fourfolds of discriminant 24 contain special codimension-two algebraic cycles of degree 6 and self-intersection 20. Such cycles may be represented by singular scrolls or del Pezzo surfaces. A discriminant 24 cubic fourfold gives rise to a twisted surface, consisting of a degree-six K3 surface and a two-torsion element of its Brauer group. We show that the cubic fourfold is rational if the Brauer class vanishes. This yields a countably-infinite collection of new examples of rational cubic fourfolds, each of codimension two in moduli.

Cubic fourfolds of discriminant 24 and rationality

Abstract

Cubic fourfolds of discriminant 24 contain special codimension-two algebraic cycles of degree 6 and self-intersection 20. Such cycles may be represented by singular scrolls or del Pezzo surfaces. A discriminant 24 cubic fourfold gives rise to a twisted surface, consisting of a degree-six K3 surface and a two-torsion element of its Brauer group. We show that the cubic fourfold is rational if the Brauer class vanishes. This yields a countably-infinite collection of new examples of rational cubic fourfolds, each of codimension two in moduli.

Paper Structure

This paper contains 19 sections, 23 theorems, 188 equations.

Table of Contents

  1. Rationality construction
  2. Sextic del Pezzo surfaces:
  3. Nodal sextic del Pezzo surfaces
  4. Equations and linear series
  5. Residuation constructions
  6. Rationality proof
  7. Hodge theory and hyperkähler geometry
  8. Statement of results
  9. Hyperkähler geometry of the variety of lines
  10. Synthetic geometry
  11. Geometry of the scrolls
  12. Relating the scrolls and del Pezzos
  13. Speculation on the correspondence
  14. Background on scrolls
  15. Twisted sheaf interpretations
  16. ...and 4 more sections

Key Result

Proposition 1.2

A nodal sextic del Pezzo surface $W$ has the following properties:

Theorems & Definitions (43)

  • Remark 1.1
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • Remark 1.4
  • proof
  • Corollary 1.5
  • Proposition 1.6
  • proof
  • Corollary 1.7
  • ...and 33 more