Toric and tropical Bertini theorems in positive characteristic

TL;DR

This paper extends classical Bertini theorems to positive characteristic settings, introducing new algebraic tools and broadening the applicability of tropical geometry results.

Contribution

It generalizes the toric Bertini theorem to positive characteristic and extends the tropical Bertini theorem to all characteristics, removing previous restrictions.

Findings

Generalization of toric Bertini theorem to positive characteristic

Extension of tropical Bertini theorem to arbitrary characteristic

Removal of characteristic dependence in tropical connectivity results

Abstract

We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. A key part of the proof is a new algebraically closed field containing the field \kk(t_1,\dots,t_d) of rational functions over an algebraically closed field \kk of prime characteristic. As a corollary, we extend the tropical Bertini theorem of Maclagan and Yu to arbitrary characteristic, which removes the characteristic dependence from the d-connectivity result for tropical varieties from that paper.