28 jan 2015 -- 16:00
Sala Seminari, DM, Pisa
Abstract.
We prove some results about the singularities of Satake compactifications of classical moduli spaces; this will give an insight into the relation among solutions of the Schottky problem in different genera. In the case of the moduli space of curves, we show that there are no stable solutions of the Schottky problem. In particular, given two inequivalent positive even unimodular quadratic forms, the difference of the associated theta series does not vanish on the Torelli locus when the genus is big enough; we are able to give an effective bound on the genus just in the rank 24 case.
On the other hand, in the hyperelliptic case, we show that the Schottky problem can be solved in a stable way. This implies that there are infinitely many pairs of quadratic forms such that the difference of the associated theta series vanishes along the hyperelliptic locus in every genus. We also provide low rank examples. Partially joint work with N. Shepherd-Barron.