Abstract：

The Hassett-Keel program seeks to offer a modular interpretation for the log minimal model program of the moduli space of stable curves. Alper, Fedorchuk, and Smyth have proposed a conjectural framework for this program by analyzing Gorenstein curve singularities under the action of G_m. Our work draws parallels between this approach and the study of the Kontsevich-Zorich conjectures pertaining to moduli spaces of Abelian differentials. We address pertinent questions in this area by employing test configurations and the Ree construction, methodologies that stem from the theory of K-stability. This research is a collaborative effort with Dawei Chen and is currently in progress.