Fermat varieties and the periods of some hypersurfaces

Details Fermat varieties and the periods of some hypersurfaces Fermat varieties and the periods of some hypersurfaces

2010-05-11

arxiv.org/abs/1005.1733v1

Mathematics

Algebraic Geometry

18 p., will appear in the Advanced Studies in Pure Mathematics 58 = Proc. Algebraic and Arithmetic Structures of Moduli Spaces

Scientific paper

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.

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