108 research outputs found
Invariants of quartic plane curves as automorphic forms
We identify the algebra of regular functions on the space of quartic
polynomials in three complex variables invariant under SL(3,C) with an algebra
of meromorphic automorphic forms on the complex 6-ball. We also discuss the
underlying geometry.Comment: 13 pages, to appear in the AMS series Contemp. Mat
Compactifications defined by arrangements II: locally symmetric varieties of type IV
We define a new class of completions of locally symmetric varieties of type
IV which interpolates between the Baily-Borel compactification and Mumford's
toric compactifications. An arithmetic arrangement in a locally symmetric
variety of type IV determines such a completion canonically. This completion
admits a natural contraction that leaves the complement of the arrangement
untouched. The resulting completion of the arrangement complement is very much
like a Baily-Borel compactification: it is the proj of an algebra of
meromorphic automorphic forms. When that complement has a moduli space
interpretation, then what we get is often a compactification obtained by means
of geometric invariant theory. We illustrate this with several examples: moduli
spaces of polarized and Enriques surfaces and the semi-universal
deformation of a triangle singularity.
We also discuss the question when a type IV arrangement is definable by an
automorphic form.Comment: The section on arrangements on tube domains has beeen expanded in
order to make a connection with a conjecture of Gritsenko and Nikulin. Also
added: a list of notation and some references. Finally some typo's corrected
and a few minor changes made in notatio
Fermat varieties and the periods of some hypersurfaces
The variety of all smooth hypersurfaces of given degree and dimension has the
Fermat hypersurface as a natural base point. In order to study the period map
for such varieties, we first determine the integral polarized Hodge structure
of the primitive cohomology of a Fermat hypersurface (as a module over the
automorphism group of the hypersurface). We then focus on the degree 3 case and
show that the period map for cubic fourfolds as analyzed by R. Laza and the
author gives complete information about the period map for cubic hypersurfaces
of lower dimension dimension. In particular, we thus recover the results of
Allcock-Carlson-Toledo on the cubic surface case.Comment: 18 p., will appear in the Advanced Studies in Pure Mathematics 58 =
Proc. Algebraic and Arithmetic Structures of Moduli Spaces, Hokkaido
University 200
Connectivity of complexes of separating curves
We prove that the separated curve complex of a closed orientable surface of
genus g is (g-3)-connected. We also obtain a connectivity property for a
separated curve complex of the open surface that is obtained by removing a
finite set from a closed one, but it is then assumed that the removed set is
endowed with a partition and that the separating curves respect that partition.
These connectivity statements have implications for the algebraic topology of
the moduli space of curves.Comment: 8 p. This version to be published in Groups, Geometry and Dynamic
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