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 moduli stacks of curves with singularities of type $A_i$ for $i \leq 3$. These are all the partial 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 compactifications via semistability with respect to suitable line bundles on $\overline{\mathcal{M}}_{g,n}(7/10)$, 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 $\overline{\mathcal{M}}_{g,n}(7/10)$, 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.