Birational Geometry of Moduli Spaces via the Essential Skeleton

Explore the intricate world of birational geometry and moduli spaces in this 53-minute lecture by Morgan Brown from the University of Miami. Delve into the concept of the essential skeleton, a polyhedral complex within the Berkovich analytification of a variety X over a valued field K. Discover how this powerful tool connects birational geometry with tropical geometry, focusing primarily on the moduli space $\mathcal{M}_{0,n}$. Examine topics such as dual complexes, space evaluations, tropicalization, and tropical compactification. Gain insights into log canonical models and their applications. Conclude with speculative discussions on higher-dimensional cases, particularly moduli spaces of hyperplane arrangements in $\mathbb{P}^2$, and participate in a Q&A session to further explore this fascinating mathematical domain.