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

1999/11/29 by Klaus Hulek, Hulek, Klaus
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #math.AG

paper · pdf · doi:10.48550/arxiv.math/9911236

14 pages

arxiv created 1999/11/29 · openalex publication_date 1999/11/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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