Taylor conditions over finite fields

TL;DR

This paper extends Poonen's Bertini theorem over finite fields to include Taylor conditions from sheaves of differentials, motivated by broader motivic results, enabling more general geometric constructions.

Contribution

It introduces a new extension of Bertini theorems incorporating Taylor conditions derived from sheaves of differentials over finite fields.

Findings

Extended Bertini theorem to Taylor conditions over finite fields

Connected Taylor conditions with motivic results of Bilu and Howe

Provided a framework for more general geometric applications

Abstract

We extend Poonen's Bertini theorem over finite fields to Taylor conditions arising from locally free quotients of the sheaf of differentials on projective space. This is motivated by a result of Bilu and Howe in the motivic setting that allows for significantly more general Taylor conditions.