Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces

We provide explicit descriptions of the generic members of Hassett’s divisors ${\mathcal{C}}_d$ for relevant $18\le d\le 38$ and for $d=44$. In doing so, we prove that ${\mathcal{C}}_d$ is unirational for $18\le d\le 38,d=44$. 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.