Explicit constructions of K3 surfaces and unirational Noether-Lefschetz divisors

TL;DR

This paper develops explicit methods to construct K3 surfaces, leading to unirational models of certain Noether-Lefschetz divisors and proving the unirationality of moduli spaces for specific genera.

Contribution

It provides explicit constructions of K3 surfaces and new proofs of the unirationality of their moduli spaces for genera 6 to 12, including genus 11.

Findings

Constructed explicit K3 surfaces for various genera.

Proved unirationality of moduli spaces for genera 6 to 12.

Identified unirational hypersurfaces in moduli spaces.

Abstract

We provide methods to construct explicit examples of $K3$ surfaces. This leads to unirational constructions of Noether--Lefschetz divisors inside the moduli space of $K3$ surfaces of genus $g$. We implement Mukai's unirationality construction of the moduli spaces of $K3$ surfaces of genus $g\in\{6,\dots, 10, 12\}$, and we also present a new constructive proof of the unirationality of the moduli space of $K3$ surfaces of genus $11$. Furthermore, we show the existence of three unirational hypersurfaces in any moduli space of $K3$ surfaces of genus $g$.