Moduli spaces of K3 surfaces and cubic fourfolds

Abstract

This thesis is concerned with the Hodge-theoretic relation between polarized K3 surfaces of degree <em>d</em> and special cubic fourfolds of discriminant <em>d</em>, as introduced by Hassett.<br /> For half of the <em>d</em>, K3 surfaces associated to cubic fourfolds come naturally in pairs. As our first main result, we prove that if (<em>S,L</em>) and (<em>S^t,L^t</em>) form such a pair of polarized K3 surfaces, then <em>S^t</em> is isomorphic to the moduli space of stable coherent sheaves on <em>S</em> with Mukai vector (3,<em>L,d</em>/6). We also explain for which <em>d</em> the Hilbert schemes Hilb^<em>n</em>(S) and Hilb^<em>n</em>(S^t) are birational.<br /> Next, we study the more general concept of associated twisted K3 surfaces. Our main contribution here is the construction of moduli spaces of polarized twisted K3 surfaces of fixed degree and order. We strengthen a theorem of Huybrechts about the existence of associated twisted K3 surfaces. We show that like in the untwisted situation, half of the time, associated twisted K3 surfaces come in pairs, and we explain how the elements of such a pair are related to each other.