The goal of birational geometry is the classification of algebraic varieties up to birational equivalence. An algebraic variety is called rational if it is birationally equivalent to the projective space. In this paper, Genki Ouchi studied the rationality problem of (complex) cubic fourfolds, that is four dimensional complex hypersurface defined by a polynomial of degree 3. Conjecturelly, very general cubic fourfolds are irrational. However, no cubic fourfold has been proven to be irrational so far. On the other hand, there are five known examples of rational cubic fourfolds. They expect that the mysterious relation between rational cubic fourfolds and K3 surfaces is a key to solve the rationality problem. There are two inconsistent conjectures about it. So they should modify one of them at least. Together with previous works, he proved that known rational cubic fourfolds satisfy both conjectures. To modify the conjectures, we have to find a new rational cubic fourfold.

Genki Ouchi
"Hilbert schemes of two points on K3 surfaces and certain rational cubic fourfolds"
arXiv: 1805.05176