Hybrid geometry of curves and their moduli spaces

https://hu-berlin.zoom.us/j/61339297016

It is now well-understood that the Deligne-Mumford compactification of the moduli spaces of curves is not large enough for capturing several limit phenomena concerning geometry of curves and their families.

The aim of the talk is to present a hybrid refinement of the Deligne-Mumford compactification, that uses the combinatorics and geometry of graphs and their minors (in the sense of Robertson-Seymour graph minor theory), and which allows to address some of these limit questions arising in the study of Riemann surfaces and their asymptotic geometry.

Among the resulting applications, I will provide a complete solution to the problem of limits of Arakelov-Bergman measures and present a refined analysis of the degenerations of Arakelov Green functions, close to the boundary of the moduli spaces.

The talk is based on joint works with Noema Nicolussi.