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 $K3$ 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