Student Algebraic Geometry: Birational rigidity and rationality

A fundamental problem in algebraic geometry is to determine which varieties are rational, that is, birational to the projective space. The Noether-Fano method for proving non-rationality is based on the idea that a Fano variety birationally equivalent to another in a nontrivial way must admit a “very singular” linear system. In this talk, we will introduce this method and prove that no smooth quartic threefold is rational.