. Università Di Pavia
On special subvarieties of Ag contained in the Torelli locus
Sala de seminarios (4to piso, lado norte de) Dpto de Matemática y Ciencia de la Computación USACH
We will speak about the Coleman-Oort conjecture on totally geodesic subvarieties of Ag, the moduli space of abelian varieties of dimension g. In order to understand the subject, we will summarize properties of Jacobians of curves, abelian varieties and the Torelli morphism.
The examples of totally geodesic subvarieties known so far are obtained as families of Jacobians of Galois coverings of curves f:C→ C'. All of them satisfy a sufficient condition, which we will denote by (∗). We will briefly explain why condition (∗) works and we will explicitly construct and study a particular example.
We will show that condition (∗) gives us a bound on the genus g' of C'. Computer calculations allow us then to say that, up to a certain genus bounded genus g of C, there are only 6 families: all of them describe Galois coverings of elliptic curves. We will quickly illustrate them.
Finally, we study the Prym maps of these families (which we will define accordingly): we will demonstrate that these families are fibered, via their Prym map, in totally geodesic curves.