Algebraic Geometry Seminar
Ben Church
Stanford University
Non-unirationality of surfaces and moduli spaces in positive characteristic.
Abstract: A variety is "unirational" if it admits a dominant rational map from projective space. For moduli spaces this amounts to an explicit “recipe” for writing down a general member of the universal family. In characteristic zero, tensor forms obstruct unirationality -- famously employed by Harris--Mumford (1982) to prove that M_g is not unirational for g > 22. In positive characteristic, unirationality behaves much wilder due to the existence of inseparable maps. Consequently, we know the (non)-unirationality of few moduli spaces in positive characteristic. I will exhibit new techniques to obstruct unirationality in positive characteristic inspired by methods used to prove hyperbolicity in complex geometry. As applications, I will present a counterexample to a 1977 conjecture of Shioda regarding the unirationality of general type surfaces and prove that many Hilbert modular varieties over positive characteristics are not unirational.
Monday March 9, 2026 at 3:00 PM in 636 SEO