The two extremal rays of some Hyper-Kähler fourfolds

TL;DR

This work classifies divisorial contractions on HK4-folds of K3^{[2]}-type with Picard rank two, showing the exceptional divisor is a conic bundle over a K3 surface and that five lattice-embedding types govern the possible conic bundles. It analyzes the movable cone via -2 and isotropic rays, employing Pell equations to describe the second extremal ray and identifying seven possible pairs of extremal rays; these results apply to Hilbert squares and, in the Fano cubic setting, reveal links to associated (twisted) K3 surfaces and Hassett loci. The paper provides explicit tables for Hilbert squares and cubic-Fano cases, clarifying when birational models correspond to S^{[2]} for a K3 surface S, when a Lagrangian fibration occurs, and how FM-partners enter the picture. It also discusses rationality consequences for cubic fourfolds, showing that several Hassett families yield rational Fano varieties of lines, with intricate connections between divisorial contractions, scrolls in Y, and the geometry of associated K3 surfaces.

Abstract

We consider projective Hyper-Kähler 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.