853 research outputs found
The Universal Kummer Threefold
The universal Kummer threefold is a 9-dimensional variety that represents the
total space of the 6-dimensional family of Kummer threefolds in 7-dimensional
projective space. We compute defining polynomials for three versions of this
family, over the Satake hypersurface, over the G\"opel variety, and over the
reflection representation of type E7. We develop classical themes such as theta
functions and Coble's quartic hypersurface using current tools from
combinatorics, geometry, and commutative algebra. Symbolic and numerical
computations for genus 3 moduli spaces appear alongside toric and tropical
methods.Comment: 50 pages; v2: added Remark 4.3, strengthened Lemma 8.3; v3: added
references and added supplementary files to sourc
Generalised Kummer constructions and Weil restrictions
We use a generalised Kummer construction to realise all but one known weight
four newforms with complex multiplication and rational Fourier coefficients in
smooth Calabi-Yau threefolds defined over the rational numbers. The Calabi-Yau
manifolds are smooth models of quotients of the Weil restrictions of elliptic
curves with CM of class number three.Comment: 12 pages; v2: added references, minor change
Arithmetic aspects of the Burkhardt quartic threefold
We show that the Burkhardt quartic threefold is rational over any field of
characteristic distinct from 3. We compute its zeta function over finite
fields. We realize one of its moduli interpretations explicitly by determining
a model for the universal genus 2 curve over it, as a double cover of the
projective line. We show that the j-planes in the Burkhardt quartic mark the
order 3 subgroups on the Abelian varieties it parametrizes, and that the Hesse
pencil on a j-plane gives rise to the universal curve as a discriminant of a
cubic genus one cover.Comment: 22 pages. Amended references to more properly credit the classical
literature (Coble in particular
Tropicalization of classical moduli spaces
The image of the complement of a hyperplane arrangement under a monomial map
can be tropicalized combinatorially using matroid theory. We apply this to
classical moduli spaces that are associated with complex reflection
arrangements. Starting from modular curves, we visit the Segre cubic, the Igusa
quartic, and moduli of marked del Pezzo surfaces of degrees 2 and 3. Our
primary example is the Burkhardt quartic, whose tropicalization is a
3-dimensional fan in 39-dimensional space. This effectuates a synthesis of
concrete and abstract approaches to tropical moduli of genus 2 curves.Comment: 33 page
Threefolds with quasi-projective universal cover
We study compact K\"ahler threefolds X with infinite fundamental group whose
universal cover can be compactified. Combining techniques from -theory,
Campana's geometric orbifolds and the minimal model program we show that this
condition imposes strong restrictions on the geometry of X. In particular we
prove that if a projective threefold with infinite fundamental group has a
quasi-projective universal cover, the latter is then isomorphic to the product
of an affine space with a simply connected manifold.Comment: 26 pages, no figure. Comments are welcom
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