A compactification of $\mathcal M_3$ via $K3$ surfaces
  S. Kond\=o defined a birational period map from the moduli space of genus three curves to a moduli space of degree four polarized K3 surfaces. In this paper we extend the period map to a surjective morphism on a suitable compactification of $\mathcal M_3$ and describe its geometry. 
Michela Artebani