Remark on polarized K3 surfaces of genus 36

TL;DR

This paper investigates polarized K3 surfaces of genus 36, showing they can be parametrized by an 18-dimensional unirational variety and are generally anticanonical sections of specific Fano 3-folds.

Contribution

It establishes a classification and parametrization of polarized K3 surfaces of genus 36 via a special lattice embedding and links them to Fano 3-folds with Gorenstein singularities.

Findings

All such K3 surfaces with a lattice embedding are parametrized by an 18-dimensional unirational variety.

A general surface is an anticanonical section of a unique Fano 3-fold with canonical Gorenstein singularities.

The study provides a geometric description of these K3 surfaces in terms of Fano 3-folds.

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.