Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces

We provide explicit descriptions of the generic members of Hassett’s divisors C_d for relevant 18 ≤ d ≤ 38 and for d D 44, which furthermore gives unirationality of these C_d. It follows as a corollary that the moduli space N_d of polarized K3 surfaces of degree d is unirational for d D 14; 26; 38. The case d 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 hyperkähler manifolds which are not birational to any moduli space of (twisted) sheaves on a K3 surface. This note is the summary of a lecture, based on the paper [Nue15], which the author gave at the summer school “Rationality problems in algebraic geometry” organized by CIME-CIRM in Levico Terme in June 2015. He would like to thank Rita Pardini and Pietro Pirola for affording him the honor of speaking and for fostering such a productive atmosphere.