#### Vol. 11, No. 4, 2017

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
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