Properties of families of projective varieties via their moduli

Date: Thursday 15 September 2022

Abstract

A celebrated conjecture of Mordell predicted that the genus of a smooth projective curve controls the density of rational solutions for the underlying polynomial equation. In the 1980s, following the footsteps of Shafarevich, Arakelov and later on Faltings observed that, thanks to Deligne-Mumford's striking construction of the moduli of high-genus curves, one can reduce Mordell's conjecture to the problem of establishing a single natural inequality over curves. This inequality, which was fundamental for Faltings' solution to Mordell's conjecture, is nowadays referred to as the Arakelov inequality. My aim in this talk is to give an overview of these interconnected themes and discuss some new advances and open questions in this area arising from recent breakthroughs in the moduli theory of higher dimensional varieties. This talk is partially based on joint work with S. Kovács.