Contractions of hyper-Kähler fourfolds and the Brauer group

Abstract

We study the geometry of exceptional loci of birational contractions of hyper-Kähler 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. Contents(1) 4,16 , EPW sextics and a special EPW sextic 39 References 43

Divisors in the moduli space of Debarre-Voisin varieties

“…Thus our Theorem 4.19 could also have been deduced by combining [KvG,Proposition 3.5] with [KvG,Theorem 4.2]. Notice finally that a crucial element for our result is Theorem 4.13, which is a particular case of [KvG,Proposition 4.6].…”

“…This follows from the fact that B • B = B • h = 1 2 , as shown in Lemma 4.15. The intersection matrix appearing in Lemma 4.9 is diagonalized in the basis H + D, D , where it becomes ( 24 0 0 −2 ); therefore the class H + D gives the contraction of the conic bundle D → S [KvG,Proposition 3.5]. Thus our Theorem 4.19 could also have been deduced by combining [KvG,Proposition 3.5] with [KvG,Theorem 4.2].…”

“…The intersection matrix appearing in Lemma 4.9 is diagonalized in the basis H + D, D , where it becomes ( 24 0 0 −2 ); therefore the class H + D gives the contraction of the conic bundle D → S [KvG,Proposition 3.5]. Thus our Theorem 4.19 could also have been deduced by combining [KvG,Proposition 3.5] with [KvG,Theorem 4.2]. Notice finally that a crucial element for our result is Theorem 4.13, which is a particular case of [KvG,Proposition 4.6].…”