Pre-talk: Moduli space of curves
I will introduce some basics on moduli space of curves and Geometric Invariant Theory (GIT). Time permitting, I will also discuss good moduli spaces.
Title: Alternative Modular Compactifications of M_{g,n} via Cluster Algebras with applications to the MMP of \overline{M}_{g,n}
Abstract:
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. By applying S- and \Theta-completeness criteria, we classify the open substacks of canonically polarized curves with nodes, cusps, and tacnodes having a proper good moduli space. 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 Q-factorialization fan of \overline{M}_{g,n}(7/10) as a cluster fan.