An application of Bertini Theorem

TL;DR

This paper extends previous results on the irreducibility of hypersurface intersections over finite fields by analyzing fixed degree cases and varying the base field, providing probability computations for linear subspace intersections.

Contribution

It introduces a new perspective by fixing degree and varying the base field, extending the understanding of intersection properties over finite fields.

Findings

Probability of linear subspace intersection with $X$ in a specified number of points.

Extension of Charles and Poonen's density results to fixed degree and varying base fields.

Analytical formulas for intersection probabilities in finite field settings.

Abstract

Given an irreducible variety $X$ over a finite field, the density of hypersurfaces of varying degree $d$ intersecting $X$ in an irreducible subvariety is $1$, by a result of Charles and Poonen. In this note, we analyse the situation fixing $d=1$ and extend the base field instead of the degree $d$. We compute the probability that a random linear subspace of the right dimension intersects $X$ in a given number of points.