Rationally isotropic exceptional projective homogeneous varieties are locally isotropic

TL;DR

This paper proves that for certain exceptional projective homogeneous varieties over specific local rings, having a rational point over the field of fractions implies the existence of a rational point over the ring itself, under certain conditions.

Contribution

It establishes a link between rational points over fields of fractions and over local rings for exceptional projective homogeneous varieties, extending known isotropy results.

Findings

X has an R-point if it has a K-point under given conditions

The result applies to varieties over local rings containing an infinite perfect field

Characteristic of the field of fractions is not 2

Abstract

Assume that R is a local regular ring containing an infinite perfect field, or that R is the local ring of a point on a smooth scheme over an infinite field. Let K be the field of fractions of R and the characteristic of K is not 2. Let X be an exceptional projective homogeneous scheme over R. We prove that in most cases the condition that X has a K-point implies that X has an R-point.