2011/03/31 by Samuel Grushevsky, Klaus Hulek · 1 citation
Mathematics · #Abelian group #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Codimension #Compactification (mathematics) #Homotopy and Cohomology in Algebraic Topology #Locus (genetics) #Mathematical analysis #Mathematics #Moduli #Moduli space #Physics #Pure mathematics #math.AG
paper · pdf · doi:10.1007/s00222-012-0377-4
v2: new section 9 on the geometry of the boundary of the locus of intermediate Jacobians of cubic threefolds. Final version to appear in Invent. Math
arxiv created 2012/01/10 · openalex publication_date 2012/01/24 · arxiv updated 2015/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 (which we show to be true for genus up to 5), we compute the class of this locus, and of is 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 classes of the closures of the locus of products of an elliptic curve and a hyperelliptic genus 3 curve, in moduli of principally polarized abelian fourfolds, and of the locus of intermediate Jacobians in genus 5. 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.