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The class of the locus of intermediate Jacobians of cubic threefolds

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Abstract

We study the locus of intermediate Jacobians of cubic threefolds within the moduli space \(\mathcal{A}_{5}\) 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 g≤5, and conjecturally for any g), we compute the class of this locus, and of its closure in the perfect cone toroidal compactification \(\mathcal{A}_{g}^{\mathrm{Perf}}\), 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 \(\mathcal{A}_{1}\times \theta_{\mathrm{null}}\) in \(\mathcal{A}_{4}^{\mathrm{Perf}}\), and for the class of the locus of intermediate Jacobians (together with the same locus of products)—in \(\mathcal{A}_{5}^{\mathrm{Perf}}\). Finally, we obtain some results on the geometry of the boundary of the locus of intermediate Jacobians of cubic threefolds in \(\mathcal{A}_{5}^{\mathrm{Perf}}\).

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.

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Authors and Affiliations

  1. Mathematics Department, Stony Brook University, Stony Brook, NY, 11790-3651, USA

    Samuel Grushevsky

  2. Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30060, Hannover, Germany

    Klaus Hulek

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  1. Samuel Grushevsky
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  2. Klaus Hulek
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Correspondence to Klaus Hulek.

Additional information

Research of the first author is supported in part by National Science Foundation under the grant DMS-10-53313.

Research of the second author is supported in part by DFG grants Hu-337/6-1 and Hu-337/6-2.

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Grushevsky, S., Hulek, K. The class of the locus of intermediate Jacobians of cubic threefolds. Invent. math. 190, 119–168 (2012). https://doi.org/10.1007/s00222-012-0377-4

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