Hassett--Keel Program in genus four

The Hassett–Keel program seeks to give a modular interpretation to the steps of the log minimal model program of ${\bar{\mathcal{M}}}_g$. The goal of this paper is to complete the Hassett–Keel program in genus four, supplementing earlier results of Casalaina-Martin–Jensen–Laza and Alper–Fedorchuk–Smyth–van der Wyck. The main tools we use are wall crossing for moduli spaces of pairs in the sense of K-stability and KSBA stability, and the recently constructed moduli spaces of boundary polarized Calabi–Yau surface pairs. We also give a construction of the hyperelliptic flip using a stackified Chow quotient which is expected to generalize to higher genus.