Remark on polarized K3 surfaces of genus 36

Abstract

Smooth primitively polarized $\mathrm{K3}$ surfaces of genus 36 are studied. It is proved that all such surfaces $S$, for which there exists an embedding $\mathrm{R} \hookrightarrow \mathrm{Pic}(S)$ of some special lattice $\mathrm{R}$ of rank 2, are parameterized up to an isomorphism by some 18-dimensional unirational algebraic variety. More precisely, it is shown that a general $S$ is an anticanonical section of a (unique) Fano 3-fold with canonical Gorenstein singularities.