Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Even nodal surfaces of K3 type

Marcello Bernardara, Enrico Fatighenti, Grzegorz Kapustka, Michał Kapustka, Laurent Manivel, Giovanni Mongardi, Fabio Tanturri

Abstract

We study Fano fourfolds of K3 type with a conic bundle structure. We construct direct geometrical links between these fourfolds and hyperKähler varieties. As a result we describe families of nodal surfaces that can be seen as generalisations of Kummer quartic surfaces. Each of these families actually arises through two families of Fano fourfolds, whose conic bundle structures are related by hyperbolic reduction.

Even nodal surfaces of K3 type

Abstract

We study Fano fourfolds of K3 type with a conic bundle structure. We construct direct geometrical links between these fourfolds and hyperKähler varieties. As a result we describe families of nodal surfaces that can be seen as generalisations of Kummer quartic surfaces. Each of these families actually arises through two families of Fano fourfolds, whose conic bundle structures are related by hyperbolic reduction.
Paper Structure (27 sections, 27 theorems, 78 equations, 2 tables)

This paper contains 27 sections, 27 theorems, 78 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Generalities on conic bundles
  3. Hyperbolic reduction
  4. A general fact about conic bundles over threefolds
  5. 16-nodal quintics and cubic fourfolds
  6. Blow-up of a cubic along a line and nodal quintics
  7. Another conic bundle with quintic discriminant
  8. Comparing the conic bundles
  9. The conic bundle structure on C-4
  10. The conic bundle structure on R-62
  11. How to connect them through hyperbolic reduction
  12. 40-Nodal sextics and Gushel--Mukai fourfolds
  13. Blow-up of GM along a degree 4 del Pezzo and nodal sextics
  14. Another conic bundle with sextic discriminant
  15. Comparing the conic bundles
  16. ...and 12 more sections

Key Result

Theorem 5

For each of the pairs $(X,X')$ of main_theorem_discriminants in the cases C4/R-62 or GM-21/K3-35, there exists a quadric threefold fibrations $Y \to B$ such that:

Theorems & Definitions (50)

  • Definition 1
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Proposition 8
  • proof
  • Proposition 9
  • proof
  • Proposition 10: hu21
  • Proposition 11
  • ...and 40 more