Howard Nuer : Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces

Posted by Howard Nuer , part of the Algebraic Geometry Seminar.

At

Sept. 6, 2017, 4 p.m.

In

SEO 427

Abstract

We provide explicit descriptions of the generic members of Hassett’s divisors $\mathcal C_d$ for relevant $18\leq d\leq 38$ and $d = 44$, thus giving unirationality of these $\mathcal C_d$. We prove as a corollary 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. We also explain the construction of what we conjecture to be a new family of irreducible symplectic manifolds which are not birational to any moduli space of (twisted) sheaves on a K3 surface. Time permitting, we explain how our results have been used by Russo and Stagliano to prove the rationality of the generic cubic fourfold in $\mathcal C_{38}$.