Bert van Geemen, Grzegorz Kapustka
Published 2021-05-17, updated 2022-12-09Version 2
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.
Comments: to appear in Advances in Mathematics
On the variety of tritangential planes to a general K3 surface of degree 6 and genus 4 in $\mP^4$
Odd order obstructions to the Hasse principle on general K3 surfaces
Real Algebraic Threefolds III: Conic Bundles
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