Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces

TL;DR

This paper demonstrates the unirationality of certain moduli spaces of special cubic fourfolds and polarized K3 surfaces, providing explicit descriptions and extending known results to new cases.

Contribution

It explicitly describes generic members of Hassett's divisors for specific degrees and proves their unirationality, including new results for degree 26 K3 surfaces.

Findings

Unirationality of $ ilde{ m C}_d$ for $18 extless d extless 38$ and $d=44$

Unirationality of $ m N_d$ for $d=14,26,38$

Explicit descriptions of generic members of Hassett's divisors

Abstract

We provide explicit descriptions of the generic members of Hassett's divisors $\mathcal C_d$ for relevant $18\leq d\leq 38$ and for $d=44$. In doing so, we prove that $\mathcal C_d$ is unirational for $18\leq d\leq 38,d=44$. As a corollary, we prove that the moduli space $\mathcal N_{d}$ of polarized K3 surfaces of degree $d$ is unirational for $d=14,26,38$. The case $d=26$ is entirely new, while the other two cases have been previously proven by Mukai.