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. 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.

number theory

Audience: researchers in the topic

Comments: pre-talk at 3pm

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UCSD number theory seminar

Series comments: Most talks are preceded by a pre-talk for graduate students and postdocs. The pre-talks start 40 minutes prior to the posted time (usually at 3:20pm Pacific) and last about 30 minutes.

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