Journal of the European Mathematical Society


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Volume 19, Issue 3, 2017, pp. 659–723
DOI: 10.4171/JEMS/678

Published online: 2017-02-15

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

Sebastian Casalaina-Martin[1], Samuel Grushevsky[2], Klaus Hulek[3] and Radu Laza[4]

(1) University of Colorado, Boulder, USA
(2) Stony Brook University, USA
(3) Universität Hannover, Germany
(4) Stony Brook University, USA

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.

Keywords: Moduli, Prym varieties, period maps, abelian varieties

Casalaina-Martin Sebastian, Grushevsky Samuel, Hulek Klaus, Laza Radu: 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), 659-723. doi: 10.4171/JEMS/678