The class of the locus of intermediate Jacobians of cubic threefolds

• 成果类型: Article
• 署名作者: Grushevsky, Samuel; Hulek, Klaus
• 署名单位: Leibniz University Hannover; State University of New York (SUNY) System; Stony Brook University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-012-0377-4
• 发表日期: 2012
• 页码: 119-168
• 关键词: moduli space; abelian-varieties; toroidal compactifications; theta-divisor; ring; SINGULARITIES; COHOMOLOGY

摘要

We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus-the locus of abelian varieties with a singular odd two-torsion point on the theta divisor. Assuming that this locus has expected codimension g (which we show to be true for ga parts per thousand currency sign5, and conjecturally for any g), we compute the class of this locus, and of its closure in the perfect cone toroidal compactification , in the Chow, homology, and the tautological ring. We work out the cases of genus up to 5 in detail, obtaining explicit expressions for the class of the closure of in , and for the class of the locus of intermediate Jacobians (together with the same locus of products)-in . Finally, we obtain some results on the geometry of the boundary of the locus of intermediate Jacobians of cubic threefolds in . In the course of our computation we also deal with various intersections of boundary divisors of a level toroidal compactification, which is of independent interest in understanding the cohomology and Chow rings of the moduli spaces.