Deformations and extensions of Gorenstein weighted projective spaces
Thomas Dedieu, Edoardo Sernesi
TL;DR
The article investigates deformations of the 14 Gorenstein weighted projective spaces of dimension 3 by examining extendability of their general anticanonical divisors $S$ through the Pinkham–Wahl framework, and relates this to deformations of the affine (and projective) cones via $T^1_{CX,-1}$. A central technical contribution is showing that for K3 surfaces with canonical singularities (and for Gorenstein WPS in their canonical embedding) one has $\alpha(X)=\dim T^1_{CX,-1}$, enabling a cone-based smoothing criterion. Using a smoothing argument for non-primitive polarizations, the authors identify eight of the fourteen WPS for which $\mathbf{P}$ deforms to a 3-fold extension of a general non-primitive polarized K3 surface, and they perform explicit Macaulay2 computations to verify the $N_2$ property and compute $\alpha$-invariants. The results yield concrete deformations and degenerations, e.g., $\mathbf{P}(1,1,4,6)$ to $\mathbf{P}(1^3,3)$, a degree-6 hypersurface in $\mathbf{P}(1^3,3,5)$, or to a Veronese embedding, illustrating the interplay between deformation theory and explicit geometric realizations. These findings advance understanding of the extendability landscape for Gorenstein WPS and provide a computable bridge to higher-dimensional Fano and K3 geometries.
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.