Deformations and extensions of Gorenstein weighted projective spaces

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.

Final version, now with Macaulay2 computations apparent

References in corpus (7)

• Compact moduli of plane curves
• On the degree of Fano threefolds with canonical Gorenstein singularities
• VersalDeformations - a package for computing versal deformations and local Hilbert schemes
• Wahl maps and extensions of canonical curves and K3 surfaces
• A remark on Fano threefolds with canonical Gorenstein singularities
• $K3$ curves with index $k>1$
• Tutorial on Tom and Jerry: the two smoothings of the anticanonical cone over PP(1,2,3)

Cited by in corpus (1)

• Action of the automorphism group on the Jacobian of Klein's quartic curve II: Invariant theta functions