On a classical correspondence between K3 surfaces II (original) (raw)
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The Moduli Space of Surfaces with K
2004
Introduction 1 1. Surfaces with 2-divisible canonical divisor 4 2. Surfaces of type II and III b. 6 3. On rings associated to curves of genus 3 9 4. The family of deformations 12 The research of the authors was performed in the realm of the DFG SCHWER-PUNKT "Globale Methode in der komplexen Geometrie", and of the EAGER EEC Project. The third author was supported by the Schwerpunkt and by P.R.I.N. 2002 "Geometria delle variet algebriche" of M.I.U.R. and is a member of G.N.S.A.G.A. of I.N.d.A.M. .
K3 surfaces and equations for Hilbert modular surfaces
Algebra & Number Theory, 2014
We outline a method to compute rational models for the Hilbert modular surfaces Y − (D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in Q( √ D), via moduli spaces of elliptic K3 surfaces with a Shioda-Inose structure. In particular, we compute equations for all thirty fundamental discriminants D with 1 < D < 100, and analyze rational points and curves on these Hilbert modular surfaces, producing examples of genus-2 curves over Q whose Jacobians have real multiplication over Q.
On a family of K3 surfaces with S4 symmetry
2011
The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is S4. There are three pairs of three-dimensional reflexive polytopes with this symmetry group, up to isomorphism. We identify a natural one-parameter family of K3 surfaces corresponding to each of these pairs, show that S4 acts symplectically on members of these families, and show that a general K3 surface in each family has Picard rank 19. The properties of two of these families have been analyzed in the literature using other methods. We compute the Picard-Fuchs equation for the third Picard rank 19 family by extending the Griffiths-Dwork technique for computing Picard-Fuchs equations to the case of semiample hypersurfaces in toric varieties. The holomorphic solutions to our Picard-Fuchs equation exhibit modularity properties known as "Mirror Moonshine"; we relate these properties to the geometric structure of our family.
N ov 2 00 5 On the Brill-Noether theory for K 3 surfaces
2008
Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c1(E),H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of coherent sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in sense of moduli is also generic in sense of Brill-Noether theory (here H is the positive generator of the Picard group of S). In case of algebraic curves such a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is sometimes called “the strong theorem of the Brill-Noether theory”. We finish by considering a number of projective examples. In particular, we construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and show the relation with th...
Bulletin of the London Mathematical Society, 2006
In a recent paper Ahlgren, Ono and Penniston described the L-series of K3 surfaces from a certain one-parameter family in terms of those of a particular family of elliptic curves. The Tate conjecture predicts the existence of a correspondence between these K3 surfaces and certain Kummer surfaces related to these elliptic curves. A geometric construction of this correspondence is given here, using results of D. Morrison on Nikulin involutions.
The Michigan Mathematical Journal, 2007
We determine all possible configurations of rational double points on complex normal algebraic K3 surfaces, and on normal supersingular K3 surfaces in characteristic p > 19.