Vari\'et\'es r\'eelles connexes non stablement rationnelles

TL;DR

This paper constructs the first known examples of certain smooth algebraic varieties over the field of real Puiseux series that are not stably rational yet have semi-algebraically connected real point sets, highlighting new phenomena in real algebraic geometry.

Contribution

It provides the first examples of smooth intersections of two quadrics in $ ext{P}_R^5$ and smooth cubic hypersurfaces in $ ext{P}_R^4$ over the field of real Puiseux series that are not stably rational but have connected real points.

Findings

Constructed examples of non-stably rational varieties with connected real points

First such examples over the field of real Puiseux series

Open problem for similar examples over the real numbers $ ext{R}$

Abstract

Let $R$ be the field of real Puiseux series. It is a real closed field. We construct the first examples of smooth intersections of two quadrics in $\mathbb{P}_R^5$ and smooth cubic hypersurfaces in $\mathbb{P}_R^4$ which are not stably rational but for which the space $X(R)$ of $R$-points is semi-algebraically connected. The question of constructing such examples over the field of real numbers $\mathbb{R}$ remains open.