Rationality of some cubic fourfolds with automorphisms

The rationality problem of cubic fourfolds is still open: it is conjectured that most of them should be irrational, but there are no explicit examples yet. In this talk, I will recall the basic notions about cubic fourfolds, their Hodge structure and moduli space, by following Hassett's viewpoint and providing some examples. Then, I will focus on some particular special cubic fourfolds admitting a nontrivial automorphism, which are proved to be rational. Therefore a question arises: are all cubic fourfolds admitting a nontrivial automorphism rational?

This talk is based on my Master's thesis at the University of Milano under the supervision of Prof. van Geemen.