Arithmetic of rationaly connected quintic 3-folds over finite and function fields

TL;DR

This paper extends classical theorems like Chevalley-Warning-Ax and Tsen to a specific geometric construction involving the blow-up of a quintic 3-fold along a line of multiplicity 3, using toric variety descriptions.

Contribution

It proves these classical theorems hold for a new class of algebraic varieties obtained via blow-ups, with proofs based on toric geometry methods.

Findings

Chevalley-Warning-Ax theorem applies to the blown-up quintic 3-fold

Tsen theorem holds for the blown-up variety

Description of the blow-up as a subvariety of a toric variety

Abstract

We prove that both classical Chevalley-Warning-Ax and Tsen theorems hold for the blowing up of a quintic 3-fold along a line of multiplicity 3. Both proofs, which are of the same spirit than the original ones, involve the description of this blowing-up as a subvariety of a toric variety.