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K3 structures from singular Fano varieties

Enrico Fatighenti

TL;DR

The paper investigates Fano varieties of K3 type (FK3) with a focus on singular degenerations that, after resolution, exhibit K3-type Hodge structures. It develops a framework around singular complete intersections of three quadrics and their blow-ups, using a $(G,\mathcal{F})$-presentation and the Cayley trick to access Hodge-theoretic data, illustrating how degenerations can yield FK3 structures. A detailed example of the K3-27 case shows how a net of quadrics produces a branch divisor whose determinant factors into a square of a linear form times a sextic, giving rise to a degree-2 K3 surface that accounts for the observed K3-structure in the resolution. These insights suggest a broader program to classify FK3-degenerations and explore their connections to hyperkähler families, potentially uncovering new K3 structures from singular degenerations.

Abstract

We survey some results obtained in our quest for Fano varieties of K3 type and discuss why exploring the singular world might be interesting for discovering new K3 structures.

K3 structures from singular Fano varieties

TL;DR

The paper investigates Fano varieties of K3 type (FK3) with a focus on singular degenerations that, after resolution, exhibit K3-type Hodge structures. It develops a framework around singular complete intersections of three quadrics and their blow-ups, using a -presentation and the Cayley trick to access Hodge-theoretic data, illustrating how degenerations can yield FK3 structures. A detailed example of the K3-27 case shows how a net of quadrics produces a branch divisor whose determinant factors into a square of a linear form times a sextic, giving rise to a degree-2 K3 surface that accounts for the observed K3-structure in the resolution. These insights suggest a broader program to classify FK3-degenerations and explore their connections to hyperkähler families, potentially uncovering new K3 structures from singular degenerations.

Abstract

We survey some results obtained in our quest for Fano varieties of K3 type and discuss why exploring the singular world might be interesting for discovering new K3 structures.

Paper Structure

This paper contains 6 sections, 1 theorem, 1 equation.

Table of Contents

  1. Introduction
  2. K3 structures from singular varieties
  3. FK3 fourfolds with singular projections
  4. The (singular) complete intersection of three quadrics
  5. Singular $(2,2)$ divisors in $\mathbb{P}^2 \times \mathbb{P}^3$ and rationality properties
  6. An example

Key Result

Proposition 2.1

The branch divisor $D \subset \Lambda$ associated to the above $\mathbb{Q}$ is the reducible union of a double line $L$ and a smooth sextic $C$.

Theorems & Definitions (2)

  • Proposition 2.1
  • proof