The Bertini irreducibility theorem for higher codimensional slices

TL;DR

This paper extends Poonen and Slavov's hyperplane slicing approach to prove bounds on the exceptional locus dimension for higher codimensional linear slices, advancing Bertini irreducibility theorems over arbitrary fields.

Contribution

It generalizes existing results by establishing bounds for higher codimension slices, broadening the applicability of Bertini theorems in algebraic geometry.

Findings

Proved bounds for the dimension of the exceptional locus in higher codimensional slices.

Extended Bertini irreducibility results to arbitrary fields with new bounds.

Generalized hyperplane slicing techniques to linear subspaces of higher codimension.

Abstract

Poonen and Slavov recently developed a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing. In this paper, we extend their work by proving an analogous bound for the dimension of the exceptional locus in the setting of linear subspaces of higher codimensions.