Igusa's modular form and the classification of Siegel modular threefolds

Details Igusa's modular form and the classification of Siegel modular threefolds Igusa's modular form and the classification of Siegel modular threefolds

1999-11-29

arxiv.org/abs/math/9911236v1

Mathematics

Algebraic Geometry

14 pages

Scientific paper

In this paper we prove two results concerning the classification of Siegel modular threefolds. Let A_{1,d}(n) be the moduli space of abelian surfaces with a (1,d)-polarization and a full level-n structure and let A_{1,d}^{lev}(n) be the space where one has fixed an additional canonical level structure. We prove that A_{1,d}(n) is of general type if (d,n)=1 and n ist at least 4. This is the best possible result which one can prove for all d simultaneously. Let p be an odd prime and assume that (p,n)=1. Then we prove that the Voronoi compactification of A_{1,p}^{lev}(n) is smooth and has ample canonical bundle if and only if n is greater than or equal to 5.

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