Abstract
We show that the degree of Gauss maps on abelian varieties is semicontinuous in families, and we study its jump loci. As an application we obtain that in the case of theta divisors this degree answers the Schottky problem. Our proof computes the degree of Gauss maps by specialization of Lagrangian cycles on the cotangent bundle. We also get similar results for the intersection cohomology of varieties with a finite morphism to an abelian variety; it follows that many components of Andreotti–Mayer loci, including the Schottky locus, are part of the stratification of the moduli space of ppav’s defined by the topological type of the theta divisor.
Highlights
The Gauss map of a hypersurface in projective space is the rational map that sends any smooth point of the hypersurface to its normal direction in the dual projective space
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]
Many Andreotti–Mayer loci such as the Schottky locus are determined over the complex numbers already by the topological type of the theta divisor
Summary
The Gauss map of a hypersurface in projective space is the rational map that sends any smooth point of the hypersurface to its normal direction in the dual projective space. In contrast to the case of projective hypersurfaces, the Gauss map for any ample divisor on an abelian variety is generically finite of degree > 1, and its degree is related to the singularities of the divisor. 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]. Many Andreotti–Mayer loci such as the Schottky locus are determined over the complex numbers already by the topological type of the theta divisor
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
