JournalsjemsVol. 19, No. 3pp. 659–723

Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with an appendix by Mathieu Dutour Sikirić)

  • Sebastian Casalaina-Martin

    University of Colorado, Boulder, USA
  • Samuel Grushevsky

    Stony Brook University, USA
  • Klaus Hulek

    Universität Hannover, Germany
  • Radu Laza

    Stony Brook University, USA
Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with an appendix by Mathieu Dutour Sikirić) cover

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Abstract

The main purpose of this paper is to present a conceptual approach to understanding the extension of the Prym map from the space of admissible double covers of stable curves to diff erent toroidal compacti cations of the moduli space of principally polarized abelian varieties. By separating the combinatorial problems from the geometric aspects we can reduce this to the computation of certain monodromy cones. In this way we not only shed new light on the extension results of Alexeev, Birkenhake, Hulek, and Vologodsky for the second Voronoi toroidal compactifi cation, but we also apply this to other toroidal compacti cations, in particular the perfect cone compacti fication, for which we obtain a combinatorial characterization of the indeterminacy locus, as well as a geometric description up to codimension six, and an explicit toroidal resolution of the Prym map up to codimension four.

Cite this article

Sebastian Casalaina-Martin, Samuel Grushevsky, Klaus Hulek, Radu Laza, Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with an appendix by Mathieu Dutour Sikirić). J. Eur. Math. Soc. 19 (2017), no. 3, pp. 659–723

DOI 10.4171/JEMS/678