Deformation of pairs of $\mathbb{P}^3$ and hypersurfaces

TL;DR

This paper investigates the deformation theory of pairs consisting of projective 3-space and hypersurfaces, revealing smoothness properties of moduli spaces and identifying boundary divisors.

Contribution

It classifies $Q$-Gorenstein degenerations of $P^3$ with canonical singularities and analyzes the moduli space structure for related threefolds.

Findings

Moduli space is smooth at pairs with degenerating threefolds having canonical singularities.

Identifies boundary divisors in the moduli space of smooth hypersurfaces.

Provides insights into the moduli of threefolds via double cover constructions.

Abstract

Motivated by DeVleming's work on moduli of surfaces in $\mathbb{P}^3$ and Chen-Hu-Jiang's work on moduli of threefolds with volume $2$ and geometric genus $4$, we study the deformation of pairs of $\mathbb{P}^3$ and hypersurfaces using the classification of $\mathbb{Q}$-Gorenstein degenerations of $\mathbb{P}^3$ with canonical singularities. We prove that if a degenerating threefold has canonical singularities, then the moduli space is smooth at the corresponding pair. Consequently, we find some boundary divisors of the moduli of smooth hypersurfaces. Finally, using the double cover method, we derive some information on the moduli space of threefolds $X$ with canonical singularities with the same volume and geometric genus as a double cover of $\mathbb{P}^3$ branched over a hypersurface.