Birational geometry of some universal families of \(n\)-pointed Fano fourfolds (Q2157473)

Cubic and Gushel-Mukai (GM) manifolds are Fano varieties with similar birational properties. It is expected that general smooth GM fourfold and cubic fourfold are non-rational but there are no known examples. On the other hand there are several known families of rational cubics and GM fourfolds. For example, cubic fourfold containing quintic del Pezzo surface is rational. The authors study some known families of cubic and GM fourfolds. They prove unirationality of \(n\)-pointed moduli space for four families of cubic fourfolds and three families of GM fourfolds. All of these families contain a surface that belongs to a unirational family of surfaces in the projective space (for cubic fourfolds) or Grassmanian (for GM fourfolds). The authors use the unirationality of the family of surfaces and the rationality of the fourfolds to prove unirationality of \(n\)-pointed moduli. The authors also prove rationality of one moduli space of rational GM fourfolds.