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We investigate the Gorenstein weighted projective spaces of dimension 3.Given such a space $\mathbf P$, our first focus is its maximal extension in itsanticanonical 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 coneand $\mathbf P$ is a linear section of $Y$. In [DS21] Thomas Dedieu and EdoardoSernesi have computed the dimension of $Y$ by cohomological computations on thecanonical curves inside $\mathbf P$. We give an explicit description of $Y$ inthe cases where it was not known. Next, we examine the general anticanonicaldivisors of $\mathbf P$. These are K3 surfaces, not necessarily primitivelypolarized. We give a geometric characterization of the curve sections in theirprimitive polarization.

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