Inventiones mathematicae

, Volume 190, Issue 1, pp 119–168 | Cite as

The class of the locus of intermediate Jacobians of cubic threefolds



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.


Modulus Space Irreducible Component Boundary Component Theta Function Abelian Variety 


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  1. 1.
    Alexeev, V.: Complete moduli in the presence of semiabelian group action. Ann. Math. (2) 155(3), 611–708 (2002) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Allcock, D., Carlson, J., Toledo, D.: The moduli space of cubic threefolds as a ball quotient. Mem. Am. Math. Soc. 209(985), xii+70 (2011) MathSciNetGoogle Scholar
  3. 3.
    Beauville, A.: Prym varieties and the Schottky problem. Invent. Math. 41(2), 149–196 (1977) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Beauville, A.: Les singularités du diviseur Θ de la jacobienne intermédiaire de l’hypersurface cubique dans P 4. In: Algebraic Threefolds, Varenna, 1981. Lecture Notes in Math., vol. 947, pp. 190–208. Springer, Berlin (1982) CrossRefGoogle Scholar
  5. 5.
    Boldsen, S.: Improved homological stability for the mapping class group with integral or twisted coefficients. Preprint arXiv:0904.3269 (2009)
  6. 6.
    Casalaina-Martin, S.: Cubic threefolds and abelian varieties of dimension five. II. Math. Z. 260(1), 115–125 (2008) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Casalaina-Martin, S.: Singularities of theta divisors in algebraic geometry. In: Curves and Abelian Varieties. Contemp. Math., vol. 465, pp. 25–43. Am. Math. Soc, Providence (2008) CrossRefGoogle Scholar
  8. 8.
    Casalaina-Martin, S., Friedman, R.: Cubic threefolds and abelian varieties of dimension five. J. Algebr. Geom. 14(2), 295–326 (2005) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Casalaina-Martin, S., Laza, R.: The moduli space of cubic threefolds via degenerations of the intermediate Jacobian. J. Reine Angew. Math. 633, 29–65 (2009) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ciliberto, C., van der Geer, G.: The moduli space of abelian varieties and the singularities of the theta divisor. In: Surveys in Differential Geometry. Surv. Differ. Geom., vol. VII, pp. 61–81. Int. Press, Somerville (2000) Google Scholar
  11. 11.
    Ciliberto, C., van der Geer, G.: Andreotti-Mayer loci and the Schottky problem. Doc. Math. 13, 453–504 (2008) MathSciNetMATHGoogle Scholar
  12. 12.
    Clemens, H., Griffiths, P.: The intermediate Jacobian of the cubic threefold. Ann. Math. 95, 281–356 (1972) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Debarre, O.: Le lieu des variétés abéliennes dont le diviseur thêta est singulier a deux composantes. Ann. Sci. Éc. Norm. Super. (4) 25(6), 687–707 (1992) MathSciNetMATHGoogle Scholar
  14. 14.
    Donagi, R.: Big Schottky. Invent. Math. 89(3), 569–599 (1987) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Donagi, R.: Non-Jacobians in the Schottky loci. Ann. Math. 126(1), 193–217 (1987) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Donagi, R.: The fibers of the Prym map. In: Curves, Jacobians, and Abelian Varieties, Amherst, MA, 1990. Contemp. Math., vol. 136, pp. 55–125. Am. Math. Soc., Providence (1992) CrossRefGoogle Scholar
  17. 17.
    Donagi, R., Smith, R.: The structure of the Prym map. Acta Math. 146(1–2), 25–102 (1981) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Ekedahl, T., van der Geer, G.: Cycles representing the top Chern class of the Hodge bundle on the moduli space of abelian varieties. Duke Math. J. 129(1), 187–199 (2005) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Erdahl, R., Ryshkov, S.: The empty sphere. II. Can. J. Math. 40(5), 1058–1073 (1988) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Erdenberger, C.: A finiteness result for Siegel modular threefolds. Ph.D. thesis, Gottfried Wilhelm Leibniz Universität Hannover, Hannover, Germany (2007) Google Scholar
  21. 21.
    Erdenberger, C., Grushevsky, S., Hulek, K.: Some intersection numbers of divisors on toroidal compactifications of \(\mathcal{A}_{g}\). J. Algebr. Geom. 19, 99–132 (2010) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Esnault, H., Viehweg, E.: Chern classes of Gauss-Manin bundles of weight 1 vanish. K-Theory 26(3), 287–305 (2002) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Faber, C.: A non-vanishing result for the tautological ring of \(\mathcal{M}_{g}\). Preprint arXiv:math/9711219 (1997)
  24. 24.
    Faber, C.: Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians. In: New Trends in Algebraic Geometry, Warwick, 1996. London Math. Soc. Lecture Note Ser., vol. 264, pp. 93–109. Cambridge Univ. Press, Cambridge (1999) CrossRefGoogle Scholar
  25. 25.
    Faber, C.: A conjectural description of the tautological ring of the moduli space of curves. In: Moduli of Curves and Abelian Varieties. Aspects Math., vol. E33, pp. 109–129. Vieweg, Braunschweig (1999) CrossRefGoogle Scholar
  26. 26.
    Fulton, W.: Intersection Theory, 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 2. Springer, Berlin (1998) MATHCrossRefGoogle Scholar
  27. 27.
    Graber, T., Vakil, R.: Relative virtual localization and vanishing of tautological classes on moduli spaces of curves. Duke Math. J. 130(1), 1–37 (2005) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Grushevsky, S., Hulek, K.: Principally polarized semiabelic varieties of torus rank up to 3, and the Andreotti-Mayer loci. Pure Appl. Math. Q., special issue in memory of Eckart Viehweg 75(4), 1309–1360 (2011). Preprint arXiv:1103.1858 MathSciNetGoogle Scholar
  29. 29.
    Grushevsky, S., Salvati Manni, R.: Gradients of odd theta functions. J. Reine Angew. Math. 573, 45–59 (2004) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Grushevsky, S., Salvati Manni, R.: Singularities of the theta divisor at points of order two. Int. Math. Res. Not. 15, Art. ID rnm045, 15 (2007) Google Scholar
  31. 31.
    Grushevsky, S., Salvati Manni, R.: The loci of abelian varieties with points of high multiplicity on the theta divisor. Geom. Dedic. 139, 233–247 (2009) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Hain, R.: The rational cohomology ring of the moduli space of abelian 3-folds. Math. Res. Lett. 9(4), 473–491 (2002) MathSciNetMATHGoogle Scholar
  33. 33.
    Harris, J.: Theta-characteristics on algebraic curves. Trans. Am. Math. Soc. 271(2), 611–638 (1982) MATHCrossRefGoogle Scholar
  34. 34.
    Hulek, K., Sankaran, G.: The nef cone of toroidal compactifications of \(\mathcal{A}_{4}\). Proc. Lond. Math. Soc. 88(3), 659–704 (2004) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Hulek, K., Tommasi, O.: Cohomology of the toroidal compactification of \(\mathcal{A}_{3}\). In: Proceedings of the Conference in honour of S. Ramanan. Contemp. Math., vol. 522, pp. 89–103 (2010) Google Scholar
  36. 36.
    Hulek, K., Kahn, C., Weintraub, S.: Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions. de Gruyter Expositions in Mathematics, vol. 12. de Gruyter, Berlin (1993) MATHCrossRefGoogle Scholar
  37. 37.
    Igusa, J.-I.: On Siegel modular forms of genus two. Am. J. Math. 84, 175–200 (1962) MathSciNetCrossRefGoogle Scholar
  38. 38.
    Igusa, J.-I.: Theta Functions. Grundlehren der Mathematischen Wissenschaften, vol. 194 (1972) MATHCrossRefGoogle Scholar
  39. 39.
    Ionel, E.-N.: Topological recursive relations in \(H^{2g}(\mathcal{M}_{g,n})\). Invent. Math. 148(3), 627–658 (2002) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Looijenga, E.: On the tautological ring of \(\mathcal{M}_{g}\). Invent. Math. 121(2), 411–419 (1995) MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Madsen, I., Weiss, M.: The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. Math. 165(3), 843–941 (2007) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Mumford, D.: Prym varieties. I. In: Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers), pp. 325–350. Academic Press, New York (1974) Google Scholar
  43. 43.
    Mumford, D.: Hirzebruch’s proportionality theorem in the noncompact case. Invent. Math. 42, 239–272 (1977) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Progr. Math., vol. 36, pp. 271–328. Birkhäuser Boston, Boston (1983) Google Scholar
  45. 45.
    Namikawa, Y.: Toroidal Compactification of Siegel Spaces. Lecture Notes in Mathematics, vol. 812. Springer, Berlin (1980) MATHGoogle Scholar
  46. 46.
    Ryškov, S., Baranovskiĭ, E.: C-types of n-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings). Proc. Steklov Inst. Math. 4, 140 (1978). Cover to cover translation of Trudy Mat. Inst. Steklov 137 (1976), Translated by R.M. Erdahl Google Scholar
  47. 47.
    Shepherd-Barron, N.: Perfect forms and the moduli space of abelian varieties. Invent. Math. 163(1), 25–45 (2006) MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Teixidor i Bigas, M.: The divisor of curves with a vanishing theta-null. Compos. Math. 66(1), 15–22 (1988) MathSciNetMATHGoogle Scholar
  49. 49.
    van der Geer, G.: The Chow ring of the moduli space of abelian threefolds. J. Algebr. Geom. 7(4), 753–770 (1998) MathSciNetMATHGoogle Scholar
  50. 50.
    van der Geer, G.: Cycles on the moduli space of Abelian varieties. In: Aspects Math., vol. E33, pp. 65–89. Vieweg, Braunschweig (1999) Google Scholar
  51. 51.
    van der Geer, G.: Corrigendum: “The Chow ring of the moduli space of abelian threefolds”. J. Algebr. Geom. 18(4), 795–796 (2009) MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Mathematics DepartmentStony Brook UniversityStony BrookUSA
  2. 2.Institut für Algebraische GeometrieLeibniz Universität HannoverHannoverGermany

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