The final log canonical model of the moduli space of stable curves of   genus four
  We describe the final log canonical model of the moduli space of stable curves of genus four. We prove that the rational map from $\bar{M}_4$ to this model contracts the Petri and the boundary divisors and flips the hyperelliptic locus. As an application, we find an exact bound on slopes of moving divisors on $\bar{M}_4$. 
Maksym Fedorchuk
math.AG