WS 2015/16 Laface – Institut für Algebraische Geometrie

Abstract Laface

The field of moduli of singular K3 surfaces

A singular K3 surface $X$ always admits a model over a number field. The absolute Galois group $\text{Gal}(\mathbb{C} / \mathbb{Q})$ acts on the set of isomorphism classes of singular K3 surfaces, and it was proven by Schütt that the orbit of $X$ under this action is in 1:1 correspondence with the genus of the transcendental lattice of $X$ . In this talk, I will introduce the notion of field of $K$ -moduli, and I will study it in two cases:

1) $K$ is the CM field of $X$ ;

2) $K = \mathbb{Q}$ .

Differences and analogies of these two settings will be highlighted. Furthermore, I will illustrate how the field of moduli varies within the moduli space of singular K3 surface. Time permitting, I will discuss some computation-oriented aspects.