Semiample Bertini theorems over finite fields

TL;DR

This paper extends Poonen's Bertini Theorem to semiample divisors over finite fields, providing new results on the existence and probability of smooth sections, with applications to counterexamples and curve distributions.

Contribution

It generalizes Bertini's theorem to semiample divisors over finite fields, introducing a probabilistic approach and new applications including counterexamples and curve analysis.

Findings

Constructed varieties that counter Bertini over finite fields.

Computed probabilities of smoothness for curves in Hirzebruch surfaces.

Analyzed distribution of points on smooth curves.

Abstract

We prove a semiample generalization of Poonen's Bertini Theorem over a finite field that implies the existence of smooth sections for wide new classes of divisors. The probability of smoothness is computed as a product of local probabilities taken over the fibers of the morphism determined by the relevant divisor. We give several applications including a negative answer to a question of Baker and Poonen by constructing a variety (in fact one of each dimension) which provides a counterexample to Bertini over finite fields in arbitrarily large projective spaces. As another application, we determine the probability of smoothness for curves in Hirzebruch surfaces, and the distribution of points on those smooth curves.