Geometry Seminar

University of Tor Vergata, Department of Mathematics

17^th of December 2024, 14:30-16:00, room D’Antoni

Alternative Modular Compactifications of $M_{g,n}$ via Cluster Algebras with applications to the MMP of ${\bar{M}}_{g,n}$

Davide Gori

Sapienza Università di Roma

We will discuss modular compactifications of $M_{g,n}$ (the moduli space of smooth curves) and their birational geometry within the framework of the Hassett-Keel program. We classify the open substacks of canonically polarized curves with nodes, cusps, and tacnodes having a proper good moduli space. Using the $S$- and $\mathrm{\Theta }$-completeness criteria, we transform the problem into a combinatorial one where compactifications and flips can be described using cluster algebra theory. This approach yields a complete description of the $\mathbb{Q}$-factorialization fan of ${\bar{M}}_{g,n}(7/10)$ as a cluster fan.

This talk is part of the activity of the MIUR Excellence Department Projects MathMod@TOV, and the PRIN 2022 Moduli Spaces and Birational Geometry and Prin PNRR 2022 Mathematical Primitives for Post Quantum Digital Signatures