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Department of Mathematics,
University of California San Diego

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Thesis Defense

Tom Grubb

UCSD

Structural and Statistical Consequences of the Closed Point Sieve

Abstract:

Poonen's Closed Point Sieve has proven to be a powerful technique for producing structural and combinatorial results for varieties over finite fields. In this talk we discuss three results which come, in part, as a consequence of this technique. First we will discuss semiample Bertini Theorems over finite fields and examine the probability with which a semiample complete intersection is smooth. Next we apply the Closed Point Sieve to compute the probability with which a high degree projective hypersurface over $\mathbb{F}_{2^k}$ is locally Frobenius split (a characteristic $p$ analog of log canonical singularities). In doing so we show that most such hypersurfaces are only mildly singular. The final part, which is based on joint work with Kiran Kedlaya and James Upton, discusses $p$-adic coefficient objects in rigid cohomology. Namely, we show (under a geometric tameness hypothesis) that the overconvergence of a Frobenius isocrystal can be detected by the restriction of that isocrystal to the collection of smooth curves on a variety.

Advisor: Kiran Kedlaya

December 6, 2022

9:00 AM

 APM 5829
Zoom Link: https://ucsd.zoom.us/j/96544659175

Research Areas

Number Theory

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