4 research outputs found
Moduli spaces of twisted K3 surfaces and cubic fourfolds
Motivated by the relation between (twisted) K3 surfaces and special cubic
fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of any
fixed degree and order. We do this by mimicking the construction of the moduli
space of untwisted polarized K3 surfaces as a quotient of a bounded symmetric
domain.Comment: 23 page
Moduli spaces of K3 surfaces and cubic fourfolds
This thesis is concerned with the Hodge-theoretic relation between polarized K3 surfaces of degree d and special cubic fourfolds of discriminant d, as introduced by Hassett. For half of the d, K3 surfaces associated to cubic fourfolds come naturally in pairs. As our first main result, we prove that if (S,L) and (St,Lt) form such a pair of polarized K3 surfaces, then St is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3,L,d/6). We also explain for which d the Hilbert schemes Hilbn(S) and Hilbn(St) are birational. 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
Kodaira dimension of moduli spaces of hyperk\"ahler varieties
We study the Kodaira dimension of moduli spaces of polarized hyperk\"ahler
varieties deformation equivalent to the Hilbert scheme of points on a K3
surface or O'Grady's ten dimensional variety. This question was studied by
Gritsenko-Hulek-Sankaran in the cases of and OG10 type when the
divisibility of the polarization is one. We generalize their results to higher
dimension and divisibility. As a main result, for almost all dimensions we
provide a lower bound on the degree such that for all higher degrees, every
component of the moduli space of polarized hyperk\"ahler varieties of
type is of general type.Comment: 56 pages. Comments are welcome