The two extremal rays of some Hyper-K\"ahler fourfolds

TL;DR

This paper classifies certain divisorial contractions on Hyper-K"ahler fourfolds deformation equivalent to Hilbert squares of K3 surfaces, identifying five types of conic bundles and detailing cases with Picard rank two, including examples for Fano varieties of cubic fourfolds.

Contribution

It provides a classification of conic bundle types arising from divisorial contractions on specific Hyper-K"ahler fourfolds and details cases with Picard rank two, including explicit examples.

Findings

Five types of conic bundles classified via lattice embeddings.

Seven cases identified for manifolds with Picard rank two and two divisorial contractions.

Four cases of conic bundles for Fano varieties of cubic fourfolds with examples provided.

Abstract

We consider projective Hyper-K\"ahler manifolds of dimension four that are deformation equivalent to Hilbert squares of K3 surfaces. In case such a manifold admits a divisorial contraction, the exceptional divisor is a conic bundle over a K3 surface. A classification of lattice embeddings implies that there are five types of such conic bundles. In case the manifold has Picard rank two and has two (birational) divisorial contractions we determine the types of these conic bundles. There are exactly seven cases. For the Fano varieties of cubic fourfolds there are only four cases and we provide examples of these.