Vol. 11, No. 4, 2017

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
The degree of the Gauss map of the theta divisor

Giulio Codogni, Samuel Grushevsky and Edoardo Sernesi

Vol. 11 (2017), No. 4, 983–1001
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