The degree of the Gauss map of the theta divisor

Abstract

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 ppav's 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.

Semicontinuity of Gauss maps and the Schottky problem

“…6 together with an analogous statement for Prym varieties. It confirms a conjecture by the first author, Grushevsky and Sernesi [11,Conjecture 1.6] who verified it for g ≤ 4 by an explicit description of the Gauss loci. As pointed out in loc.…”

“…We show that this degree is lower semicontinuous in families, and we study its jump loci. As an application we get that in the moduli space of principally polarized abelian varieties, the degree of the Gauss map refines the Andreotti-Mayer stratification and answers the Schottky problem as conjectured in [11]. We work over an algebraically closed field k with char(k) = 0.…”

“…However, in general the degree is neither upper nor lower semi-continuous, as the following variation of [11,Ex. 2.3] shows: Example 4.2 Let S be a smooth affine curve with two marked points s ± ∈ S(k) and fix positive integers n ± ≤ n < 27.…”

Hodge Ideals

“…(1) The last assertion in Theorem 29.5 becomes better than that given by Theorem 29.2 for g very large. Grushevsky (together with Codogni and Sernesi [CGS16]), and independently Lazarsfeld, have communicated to us that they can show a similar, but slightly stronger statement, using methods from intersection theory. In [CGS16] it is shown that if m is the multiplicity of an isolated point on Θ, then m(m − 1) g−1 ≤ g!…”

The Gauss map and secants of the Kummer variety

“…Recall that the Gauss map is a dominant rational morphism whose domain is the smooth locus . A basic reference is [, Section 4.4]; some recent research papers on the Gauss map and their generalizations are .…”