Structural and Statistical Consequences of the Closed Point Sieve

Poonen’s Closed Point Sieve has proven to be a powerful technique for producingstructural and combinatorial results for varieties over finite fields. In this thesis we will discuss three results which come, in part, as a consequence of this technique. First we will discuss semiample Bertini Theorems over finite fields, wherein we examine the probability with which a semiample complete intersection is smooth. In doing so we generalize work of Bucur and Kedlaya to the semiample setting of Erman and Wood. In the next chapter we apply the Closed Point Sieve to compute the probability with which a high degree projective hypersurface over F2 is locally Frobenius split (a characteristic p analog of log canonical singularities). This probability approaches 1 as the dimension of the ambient projective space grows, showing that “most” projective hypersurfaces over F2 are only mildly singular. The final chapter, which is based on joint work with Kiran Kedlaya and James Upton, discusses an application of Bertini Theorems over finite fields to the topic of 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