3:45: Modular, log canonical, and tropical compactificationsJenia TevelevThe moduli space of stable curves of Grothendieck and Knudsen has a straightforward generalization in all dimensions: the moduli space of stable pairs of Kollar, Shepherd-Barron, and Alexeev. Its construction relies on Mori theory and its boundary is very difficult to describe. Using the classical example of a cubic surface with 27 lines, I'll describe joint work with Hacking and Keel where we construct the moduli space of stable Del Pezzo surfaces and describe its boundary using non-archimedean amoebas. This is one instance of the beautiful connection between Mori theory and tropical geometry. |
PDF notes, courtesy of Bjorn Poonen
5:00: Abel maps for stable curvesEduardo EstevesThe Abel map embeds a nonsingular projective curve in a projective algebraic group, the so-called Jacobian variety of the curve. Using the group structure we can consider higher versions of the Abel map, which carry a lot of information about the projective geometry of the curve. If the curve varies in a family, so do its Jacobian variety and the Abel map. So it is natural to ask what happens when a curve degenerates to a singular curve, and more specifically, to a Deligne-Mumford stable curve. We will see in this talk how to construct an analogue of the Abel map that "nearly" embeds a stable curve in a generalization of the Jacobian variety. This is joint work with Caporaso (Roma 3) and Coelho (IMPA). |
PDF notes, courtesy of Bjorn Poonen