A variety is unirational if it admits a dominant rational map from projective space. In characteristic zero, global tensor forms obstruct unirationality. This is the principle behind the Harris–Mumford theorem (1982): M_g is of general type, and a fortiori not unirational, for g large. In positive characteristic the picture is far wilder, owing to the existence of inseparable maps, and as a result the unirationality of only a handful of moduli spaces is understood.

I will introduce new techniques for obstructing unirationality in positive characteristic, inspired by methods for proving hyperbolicity in complex geometry. As applications, I give a counterexample to Shioda's 1977 conjecture that a simply connected surface in positive characteristic is unirational if and only if it is supersingular. I also show that many Hilbert modular varieties in positive characteristic are not unirational or even covered by rational or elliptic curves.