What question did this study set out to answer?

This research aims to classify conic bundles associated with hyper-Kähler fourfolds that have deformation equivalent structures to Hilbert squares of K3 surfaces.

July 11, 2026Open Access

The Two Extremal Rays of Some Hyper–Kähler Fourfolds

FGFederica GalluzziCollegio Carlo Alberto BGBert van GeemenState University of New York

Key Points

This research aims to classify conic bundles associated with hyper-Kähler fourfolds that have deformation equivalent structures to Hilbert squares of K3 surfaces.
Analyzed projective hyper-Kähler manifolds of dimension 4 for divisorial contractions.
Employed classification of lattice embeddings to identify types of conic bundles.
Investigated cases specifically for manifolds with Picard rank 2 and multiple divisorial contractions.
Identified five distinct types of conic bundles related to the exceptional divisor in hyper-Kähler manifolds.
Determined exactly seven cases of conic bundles for manifolds with Picard rank 2 and two divisorial contractions.
Found only four cases for the Fano varieties of lines of cubic fourfolds, providing specific examples.

Abstract

ABSTRACT We consider projective Hyper–Kähler manifolds of dimension 4 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 2 and has two (birational) divisorial contractions, we determine the types of these conic bundles. There are exactly seven cases. For the Fano varieties of lines of cubic fourfolds there are only four cases and we provide examples of these.

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