Deformations and extensions of Gorenstein weighted projective spaces

TL;DR

This paper investigates the deformations of 14 Gorenstein weighted projective spaces of dimension 3, identifying cases where they extend to higher dimensions and analyzing their algebraic properties.

Contribution

It provides the first comprehensive deformation analysis of these spaces, including explicit counts of extendability and properties like N2.

Findings

8 out of 14 spaces deform to 3D extensions of K3 surfaces

Each space satisfies property N2 in its anticanonical model

Computed the deformation space of the cone over each space

Abstract

We study the existence of deformations of all $14$ Gorenstein weighted projective spaces $\mathbf P$ of dimension $3$ by computing the number of times their general anticanonical divisors are extendable. In favorable cases (8 out of 14), we find that $\mathbf P$ deforms to a $3$-dimensional extension of a general non-primitive polarized $K3$ surface. On our way we show that each such $\mathbf P$ in its anticanonical model satisfies property $N_2$, and we compute the deformation space of the cone over $\mathbf P$. This gives as a byproduct the exact number of times $\mathbf P$ is extendable.