Thursday 17 January 2019, 16:00-17:00 in HG00.310
Otto Overkamp (Hannover)
Kulikov models of Kummer surfaces
Let \(K\) be a complete discrete valuation field with algebraically closed residue field \(k\); we assume that the characteristic of \(k\) is different from \(2\). Let \(X\) be a K3 surface over \(K\), i.e. a smooth, projective, and geometrically integral algebraic surface over \(K\) with trivial canonical sheaf and trivial \(H^1(X, \mathcal O_X)\). In general, it is an open question whether we can find a finite extension of \(K \) such that there exists a semistable model of \(X\) over the ring of integers of that finite extension, even if we allow the model to be an algebraic space rather than a scheme. I shall explain how the question can be answered affirmatively if \(X\) is the Kummer surface associated with some Abelian surface over \(K\). In fact, we can even show that the models we construct are schemes, and that their relative canonical sheaf vanishes (i.e., the models we construct are so-called Kulikov models). Time permitting, I shall say a few words about the general theory of Kulikov models.