Compact Moduli of Surfaces in Three-Dimensional Projective Space

Abstract

The main goal of this paper is to construct a compactification of the moduli space of degree $d \ge 5$ hypersurfaces in $\mathbb{P}^3$, i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth hypersurfaces in $\mathbb{P}^3$ and whose boundary points correspond to degenerations of such hypersurfaces. Following a trail blazed by numerous others (see, for example, work of Koll\'ar, Shepherd-Barron, Alexeev, and Hacking), we consider a hypersurface $D$ in $\mathbb{P}^3$ as a pair $(\mathbb{P}^3, D)$ satisfying certain properties. We find a modular compactification of such pairs and use their properties to classify the pairs on the boundary to the moduli space.