706 research outputs found
Low-dimensional Singularities with Free Divisors as Discriminants
We present versal complex analytic families, over a smooth base and of fibre
dimension zero, one, or two, where the discriminant constitutes a free divisor.
These families include finite flat maps, versal deformations of reduced curve
singularities, and versal deformations of Gorenstein surface singularities in
C^5. It is shown that such free divisors often admit a "fast normalization",
obtained by a single application of the Grauert-Remmert normalization
algorithm. For a particular Gorenstein surface singularity in C^5, namely the
simple elliptic singularity of type \tilde A_4, we exhibit an explicit
discriminant matrix and show that the slice of the discriminant for a fixed
j-invariant is the cone over the dual variety of an elliptic curve.Comment: 29 pages, misprints and references correcte
Orbit Parametrizations for K3 Surfaces
We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of
representations of algebraic groups. In particular, over an algebraically
closed field of characteristic 0, we show that in many cases, the nondegenerate
orbits of a representation are in bijection with K3 surfaces (up to suitable
equivalence) whose N\'eron-Severi lattice contains a given lattice. An
immediate consequence is that the corresponding moduli spaces of these
lattice-polarized K3 surfaces are all unirational. Our constructions also
produce many fixed-point-free automorphisms of positive entropy on K3 surfaces
in various families associated to these representations, giving a natural
extension of recent work of Oguiso.Comment: 83 pages; to appear in Forum of Mathematics, Sigm
The moduli space of cubic threefolds with a non-Eckardt type involution via intermediate Jacobians
There are two types of involutions on a cubic threefold: the Eckardt type
(which has been studied by the first named and the third named authors) and the
non-Eckardt type. Here we study cubic threefolds with a non-Eckardt type
involution, whose fixed locus consists of a line and a cubic curve.
Specifically, we consider the period map sending a cubic threefold with a
non-Eckardt type involution to the invariant part of the intermediate Jacobian.
The main result is that the global Torelli Theorem holds for the period map. To
prove the theorem, we project the cubic threefold from the pointwise fixed line
and exhibit the invariant part of the intermediate Jacobian as a Prym variety
of a (pseudo-)double cover of stable curves. The proof relies on a result of
Ikeda and Naranjo-Ortega on the injectivity of the related Prym map. We also
describe the invariant part of the intermediate Jacobian via the projection
from a general invariant line and show that the two descriptions are related by
the bigonal construction.Comment: 27 pages, final version, to appear in IMR
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