Smooth hypersurface sections containing a given subscheme over a finite field

TL;DR

This paper proves a variant of Bertini's theorem over finite fields, showing that a positive fraction of hypersurfaces containing a given subscheme intersect a smooth variety smoothly, under certain dimension conditions.

Contribution

It introduces a new approach using the closed point sieve to analyze smooth hypersurface sections over finite fields, extending Bertini-type results.

Findings

Positive fraction of hypersurfaces contain Z and intersect X smoothly when m>2l

Provides explicit computation of the fraction of such hypersurfaces

Extends classical Bertini theorems to finite field settings

Abstract

We use the "closed point sieve" to prove a variant of a Bertini theorem over finite fields. Specifically, given a smooth quasi-projective subscheme X of P^n of dimension m over F_q, and a closed subscheme Z in P^n such that Z intersect X is smooth of dimension l, we compute the fraction of homogeneous polynomials vanishing on Z that cut out a smooth subvariety of X. The fraction is positive if m>2l.