Rationality of the universal K3 surface of genus 8

TL;DR

This paper proves the rationality of the universal family of polarized K3 surfaces of degree 14 by relating it to a moduli space of cubic fourfolds and quartic scrolls, establishing its rationality.

Contribution

It demonstrates the rationality of the universal K3 surface family of genus 8 through a novel identification with a moduli space involving cubic fourfolds and quartic scrolls.

Findings

Universal K3 surface family of genus 8 is rational.

Moduli space of cubic fourfolds with quartic scrolls is rational.

The approach uses a structure of a projective bundle over a stably rational variety.

Abstract

The aim of the present paper is to prove the rationality of the universal family of polarized $ K3 $ surfaces of degree 14. This is achieved by identifying it with the moduli space of cubic fourfolds plus the data of a quartic scroll. The last moduli space is finally proved to be rational since it has a natural structure of $\mathbb P^n$-bundle over a $ k $-stably rational variety with $k \leq n$.