We investigate the Gorenstein weighted projective spaces of dimension 3. Given such a space \(\textbf{P}\) , our first focus is its maximal extension in its anticanonical model \(\textbf{P}\subset \textbf{P}^{g+1}\) , i.e., the variety \(Y\subset \textbf{P}^{g+1+r}\) of largest dimension such that Y is not a cone and \(\textbf{P}\) is a linear section of Y. In Dedieu - Sernesi (The Art of Doing Algebraic Geometry) Thomas Dedieu and Edoardo Sernesi have computed the dimension of Y by cohomological computations on the canonical curves inside \(\textbf{P}\) . We give an explicit description of Y in the cases where it was not known. Next, we examine the general anticanonical divisors of \(\textbf{P}\) . These are K3 surfaces, not necessarily primitively polarized. We give a geometric characterization of the curve sections in their primitive polarization.
