Extension of Gorenstein weighted projective 3-spaces and characterization of the primitive curves of their surface sections

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.