Measures of Irrationality for Hypersurfaces of Large Degree

Given an n-dimensional smooth hypersurface X of degree d in projective space, it is elementary that X is irrational when d > n+1, but it is interesting to ask "how irrational" such a hypersurface can be. We discuss various measures of irrationality, and show that they are governed by birational positivity properties of the canonical bundle. Among other things, we prove a conjecture of Bastianelli, Cortina and De Poi concerning the least degree with which X can be expressed as a rational covering of projective space. Time permitting, I will also discuss some open problems, and some further results of Bastianelli, Clilberto, Flamini et al computing related invariants for hypersurfaces. This is joint work with Bastianelli, De Poi, Ein and Ullery.