The degree of the Gauss map of the theta divisor

Codogni, Giulio; Grushevsky, Samuel; Sernesi, Edoardo

doi:10.2140/ant.2017.11.983

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The degree of the Gauss map of the theta divisor

Giulio Codogni, Samuel Grushevsky and Edoardo Sernesi

Vol. 11 (2017), No. 4, 983–1001

DOI: 10.2140/ant.2017.11.983

Abstract

We study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. Thanks to this analysis, we obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus. We spell out this bound in several examples, and we use it to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppavs by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension.

Keywords

Gauss map, principally polarised abelian varieties, Schottky problem, V-cycles, excess intersection formula

Mathematical Subject Classification 2010

Primary: 14K10

Secondary: 14C17, 14H42

Milestones

Received: 17 August 2016

Revised: 10 January 2017

Accepted: 11 February 2017

Published: 18 June 2017

Authors

Giulio Codogni
Dipartimento di Matematica e Fisica Università Roma Tre Largo San Leonardo Murialdo I-00146 Rome Italy

Samuel Grushevsky
Mathematics Department Stony Brook University Stony Brook, NY 11794-3651 United States

Edoardo Sernesi
Dipartimento di Matematica e Fisica Università Roma Tre Largo San Leonardo Murialdo I-00146 Roma Italy

Vol. 11, No. 4, 2017