2,293 research outputs found
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
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
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
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
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