Quasitriviality of the Forms of Segre Varieties

TL;DR

This paper proves that certain algebraic forms of Segre varieties are rational if they have a rational point, using Galois-invariant projections, and extends results to hyperplane sections.

Contribution

It introduces a method to establish the rationality and quasitriviality of forms of Segre varieties via Galois-invariant birational projections.

Findings

Proves rationality of forms of Segre varieties with a rational point.

Establishes quasitriviality of forms of hyperplane sections of Segre varieties.

Uses Galois-invariant projections as a key technique.

Abstract

We prove the rationality of a $\k$-form $X$ of the product $S$ of projective spaces provided the existence of a $\k$-point on $X$. The method of the proof is to find a Galois-invariant birational projection of $S$ to the projective space. This method also allows to prove the quasitriviality of the forms of the hyperplane sections of some Segre varieties.