Seminar Algebraic Geometry

I will explain how, from a log scheme, one can extract a combinatorial object which retains a lot of its geometric features, the tropicalization. We will see how tropicalizations help reduce potentially difficuly algebro-geometric questions to tractable combinatorics. Then, I will discuss some of the great successes of logarithmic moduli theory: the logarithmic compactifications of the universal Jacobian, of the moduli space of abelian varieties, and their tropicalizations.

The classical Torelli theorem states that a smooth projective curve can be recovered from its principally polarized Jacobian. A natural generalization is the Prym-Torelli problem, which asks whether a covering of curves can be recovered from its associated polarized Prym variety. I will start with a brief introduction to Prym theory and then focus on the Prym map for étale cyclic covers of degree four of hyperelliptic curves. I will show that, for genus at least three, the Prym map is injective on one of the irreducible components. For genus two, I will describe the fibers of the Prym map, which, unlike in higher genus, are isomorphic to the punctured complex line, apart from one exceptional nonreduced fiber.