Moduli of surfaces in $\mathbb{P}^3$

Abstract

The goal of this paper is to construct a compactification of the moduli space of degree $d \ge 5$ surfaces in $\mathbb{P}^3$, i.e. a parameter space whose interior points correspond to (equivalence classes of) smooth surfaces in $\mathbb{P}^3$ and whose boundary points correspond to degenerations of such surfaces. We study a more general problem and consider a divisor $D$ on a Fano variety $Z$ as a pair $(Z, D)$ satisfying certain properties. We find a modular compactification of such pairs and, in the case of $Z = \mathbb{P}^3$ and $D$ a surface, use their properties to classify the pairs on the boundary of the moduli space.