Contractions of hyper-K\"ahler fourfolds and the Brauer group

TL;DR

This paper investigates the geometry of certain contractions in hyper-K"ahler fourfolds of K3$^{[2]}$-type, focusing on their conic bundle structures and Brauer group classes, using twisted sheaves and Heegner divisors.

Contribution

It characterizes the classes of conic bundle loci in the Brauer group of hyper-K"ahler fourfolds and relates them to twisted sheaves and Heegner divisors, revealing three classes of order two elements.

Findings

Conic bundle loci are classified in the Brauer group.

Exactly two classes of order two Brauer elements are represented by conic bundles.

Examples of such conic bundles are provided.

Abstract

We study the geometry of exceptional loci of birational contractions of hyper-K\"ahler fourfolds that are of K3$^{[2]}$-type. These loci are conic bundles over K3 surfaces and we determine their classes in the Brauer group. For this we use the results on twisted sheaves on K3 surfaces, on contractions and on the corresponding Heegner divisors. For a general K3 surface of fixed degree there are three (T-equivalence) classes of order two Brauer group elements. The elements in exactly two of these classes are represented by conic bundles on such fourfolds. We also discuss various examples of such conic bundles.