Extension of Gorenstein weighted projective 3-spaces and characterization of the primitive curves of their surface sections
Bruno Dewer
TL;DR
This work advances the understanding of extendability for Gorenstein weighted projective 3-spaces by detailing maximal extensions in the anticanonical framework and providing geometric descriptions of primitive-polarization curves on their K3 surface sections. It builds on prior cohomological results for canonical curves to compute extension dimensions, identifies the six nongeneral cases where maximal extensions exist, and furnishes explicit ambient-space constructions (via Veronese embeddings and birational models) that realize these extensions. The authors also characterize, for each nongeneral case, the primitive polarization curves C in terms of plane models, hyperelliptic/trigonal structures, or blowups, linking the curve geometry directly to the ambient weighted projective setup. Overall, the paper contributes concrete, model-driven descriptions of extensions and primitive-polarization curves for the 3-dimensional Gorenstein weighted projective spaces, enabling further investigation of their moduli and embedding properties.
Abstract
We investigate the Gorenstein weighted projective spaces of dimension 3. Given such a space $\mathbf P$, our first focus is its maximal extension in its anticanonical model $\mathbf P \subset \mathbf P^{g+1}$, i.e., the variety $Y\subset \mathbf P^{g+1+r}$ of largest dimension such that $Y$ is not a cone and $\mathbf P$ is a linear section of $Y$. In [DS23] Thomas Dedieu and Edoardo Sernesi have computed the dimension of $Y$ by cohomological computations on the canonical curves inside $\mathbf P$. We give an explicit description of $Y$ in the cases where it was not known. Next, we examine the general anticanonical divisors of $\mathbf P$. These are K3 surfaces, not necessarily primitively polarized. We give a geometric characterization of the curve sections in their primitive polarization.