13:30 - 14:30 Amira Tlemsani (Leiden)
15:00 - 16:00 Ruben de Preter (Antwerp)
16:30 - 17:30 Zhelun Chen (Leiden)
Amira Tlemsani , DR cycles in higher relative dimensions
The double ramification cycle of a line bundle on a smooth curve over a scheme S is a cycle in S roughly corresponding to the set of points where the restriction of the line bundle to the fiber is trivial. The problem of computing these cycles (in the more general setting of stable curves) was first posed by Eliashberg in 2001, and has gained a lot of popularity since then. In this talk we will define and compute DR cycles for smooth projective schemes of any relative dimension, by taking inspiration from a suitable proof of the DR cycle formula for smooth curves.
Ruben de Preter, The Kodaira dimension of M_{3,n}
For relatively low genus g the moduli space M_g is uniruled and therefore has negative Kodaira dimension. When g grows, the birational geometry of M_g becomes more complicated and for g at least 22 the space M_g has maximal Kodaira dimension. More generally, for g at least 2, the Kodaira dimension of the moduli spaces of pointed curves M_{g,n} is maximal in all but finitely many cases. For genus g=3, this result is new. In this talk I will explain the main ideas behind the proof of the result in genus 3.
Zhelun Chen, Degenerations of the archimedean height pairing.
Abstract: Let <,> denote the archimedean height pairing on a smooth projective complex variety. Given two flat families of algebraic cycles (Z_t) and (W_t) moving in a smooth projective family X/S of complex varieties parametrized by a curve S, we are interested in the asymptotic behaviour of the function h(t)=<Z_t,W_t> as t approaches to a boundary point of S, over which the family X/S degenerates. We conjecture that the asymptotics is controlled by the geometric height pairing for the generic fibers of (Z_t) and (W_t), and verify the case where the cycles are algebraically trivial, generalizing an earlier result of Holmes–de Jong. The key inputs are Künnemann's theorem of the height pairing of algebraically trivial cycles and the recent framework of Achter–Casalaina-Martin–Vial on relative algebraic intermediate Jacobians.