|Jul 02, 2018
Smooth cubic surfaces have been known to be rational since 19th century. However, Clemens and Griffiths proved in 1972 that smooth cubic threefolds are not rational by considering their intermediate Jacobians. The rationality of smooth cubic fourfolds is an open question and seems to be more difficult. There are several approaches in this direction: Hassett's approach using Hodge theory and Kuznetsov's approach using derived categories. In this talk, I will explain some examples of rational cubic fourfolds including Pfaffian cubics, Hodge theory of cubic fourfolds, and special cubic fourfolds having associated K3 surfaces.