Homotopy Rational Points of Brauer-Severi Varieties

TL;DR

This paper investigates homotopy rational points of Brauer-Severi varieties over characteristic zero fields, exploring conditions for splitting and the role of Chern classes, with results on varieties over p-adic fields.

Contribution

It establishes the relationship between homotopy rational points and splitting of Brauer-Severi varieties, and identifies the obstruction via Chern classes, extending understanding in algebraic geometry.

Findings

A Brauer-Severi variety with a homotopy rational point splits.

Open subvarieties of twisted hyperplane arrangements satisfy the splitting property.

Every Brauer-Severi variety over a p-adic field with p-primary period admits a homotopy rational point.

Abstract

We study homotopy rational points of Brauer-Severi varieties over fields of characteristic zero. We are particularly interested if a Brauer-Severi variety admitting a homotopy rational point splits. The analogue statement turns out to be true for open subvarieties of twisted hyperplane arrangements. In the complete case, the obstruction turns out to be the pullback of the Chern class of a generator of the Picard group along the homotopy rational point. Moreover, we will show that every Brauer-Severi variety over a $p$-adic field with $p$-primary period admits a homotopy rational point.