Rationality of the universal K3 surface of genus 8

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 $k \leq n$.