Thesis title: Alternative Compactifications of M_{g,n} via Cluster Algebras and their Birational Geometry
We construct new compactifications of $M_{g,n}$ as good moduli spaces of stacks of curves with $A_i$-singularities for $i \leq 3$. These are all the partial $\mathbb{Q}$-factorizations of $\overline{M}_{g,n}(7/10)$, the space appearing in the first flip of the Hassett-Keel program, providing a new instance of the modularity principle for the minimal model program of $\overline{M}_{g,n}$.
We study the stack of curves with ample log dualizing sheaf and singularities of the above type, establishing a characterization of open substacks that admit a proper good moduli space. We then recover the new compactifications as the good moduli space of semistable loci with respect to suitable line bundles on $\M_{g,n}(7/10)$, the stack of curves appearing in the first flip of the Hassett-Keel program.
Our approach develops a framework for studying semistability with respect to line bundles, revealing a wall-crossing phenomenon in a quotient of the Picard group; in the case of $\M{g,n}(7/10)$, this wall-crossing is given by the cluster fan of certain finite-type cluster algebras.
This work extends the results and answers open questions in a paper by Codogni, Tasin, and Viviani.