Log minimal model program for the moduli space of stable curves of genus three

In this paper, we completely work out the log minimal model program for the moduli space of stable curves of genus three. We employ a rational multiple $\alpha\delta$ of the divisor $\delta$ of singular curves as the boundary divisor, construct the log canonical model for the pair $(\bar{\mathcal M}_3, \alpha\delta)$ using geometric invariant theory as we vary $\alpha$ from one to zero, and give a modular interpretation of each log canonical model and the birational maps between them. By using the modular description, we are able to identify all but one log canonical models with existing compactifications of $M_3$, some new and others classical, while the exception gives a new modular compactification of $M_3$.