The Geometry and Moduli of K3 Surfaces

These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of polarized K3 surfaces, studying their moduli, degenerations and the compactification problem. This theory is then further enhanced to a discussion of lattice polarized K3 surfaces, which provide a rich source of explicit examples, including a large class of lattice polarizations coming from elliptic fibrations. Finally, we conclude by discussing the ample and Kahler cones of K3 surfaces, and give some of their applications.Comment: 34 pages, 2 figures. (R. Laza, M. Schutt and N. Yui, eds.

Kahlerian K3 surfaces and Niemeier lattices

Using results (especially see Remark 1.14.7) of our paper "Integral symmetric bilinear forms and some of their applications", 1979, we clarify relation between Kahlerian K3 surfaces and Niemeier lattices. We want to emphasise that all twenty four Niemeier lattices are important for K3 surfaces, not only the one which is related to the Mathieu group.Comment: Var7: 88 pages. We added last case

The Kodaira dimension of the moduli of K3 surfaces

The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46,50,54,58,60.Comment: 47 page

Del Pezzo surfaces with 1/3(1,1) points

We classify del Pezzo surfaces with 1/3(1,1) points in 29 qG-deformation families grouped into six unprojection cascades (this overlaps with work of Fujita and Yasutake), we tabulate their biregular invariants, we give good model constructions for surfaces in all families as degeneracy loci in rep quotient varieties and we prove that precisely 26 families admit qG-degenerations to toric surfaces. This work is part of a program to study mirror symmetry for orbifold del Pezzo surfaces.Comment: 42 pages. v2: model construction added of last remaining surface, minor corrections, minor changes to presentation, references adde

The rationality of the moduli spaces of Coble surfaces and of nodal Enriques surfaces

We prove the rationality of the coarse moduli spaces of Coble surfaces and of nodal Enriques surfaces over the field of complex numbers.Comment: 15 page

Symplectic involutions on deformations of K3^[2]

Let X be a Hyperk\"{a}hler variety deformation equivalent to the Hilbert square on a K3 surface and let f be an involution preserving the symplectic form. We prove that the fixed locus of f consists of 28 isolated points and 1 K3 surface, moreover the anti-invariant lattice of the induced involution on H^2(X,Z) is isomorphic to E_8(-2). Finally we prove that any couple consisting of one such variety and a symplectic involution on it can be deformed into a couple consisting of the Hilbert square of a K3 surface and the involution induced by a Nikulin involution on the K3 surface.Comment: Final version, to appear on Central European Journal of Mathematic

Classification of K3-surfaces with involution and maximal symplectic symmetry

K3-surfaces with antisymplectic involution and compatible symplectic actions of finite groups are considered. In this situation actions of large finite groups of symplectic transformations are shown to arise via double covers of Del Pezzo surfaces. A complete classification of K3-surfaces with maximal symplectic symmetry is obtained.Comment: 26 pages; final publication available at http://www.springerlink.co

Structure and stresses in a system of two mechanical twins in titanium

In the work we have presented the results of experimental studies and mathematical modeling for the processes of the structure formation in a transition zone of wedge-type twins system in commercially pure titanium. The process of interaction of structure defects with twinning dislocations during the formation of a wedge-type twin was taken into consideration. It is shown that the interaction alters the stress maximum in vicinity of boundaries in the system two wedge-type twi

On the 0-dimensional cusps of the Kahler moduli of a K3 surface

Let S be a projective K3 surface. It is proved that the 0-dimensional cusps of the Kahler moduli of S are in one-to-one correspondence with the twisted Fourier-Mukai partners of S. This leads to a counting formula for the 0-dimensional cusps of the Kahler moduli. Applications to rational maps between K3 surfaces with large Picard numbers are given. When the Picard number of S is 1, the bijective correspondence is calculated explicitly.Comment: 24page

Lagrangian fibrations of holomorphic-symplectic varieties of K3^[n]-type

Let X be a compact Kahler holomorphic-symplectic manifold, which is deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface. Let L be a nef line-bundle on X, such that the 2n-th power of c_1(L) vanishes and c_1(L) is primitive. Assume that the two dimensional subspace H^{2,0}(X) + H^{0,2}(X), of the second cohomology of X with complex coefficients, intersects trivially the integral cohomology. We prove that the linear system of L is base point free and it induces a Lagrangian fibration on X. In particular, the line-bundle L is effective. A determination of the semi-group of effective divisor classes on X follows, when X is projective. For a generic such pair (X,L), not necessarily projective, we show that X is bimeromorphic to a Tate-Shafarevich twist of a moduli space of stable torsion sheaves, each with pure one dimensional support, on a projective K3 surface.Comment: 34 pages. v3: Reference [Mat5] and Remark 1.8 added. Incorporated improvement to the exposition and corrected typos according to the referees suggestions. To appear in the proceedings of the conference Algebraic and Complex Geometry, Hannover 201