Certaines fibrations en surfaces quadriques r\'eelles

TL;DR

This paper investigates the stable rationality of real threefolds fibred into quadric surfaces over the real projective line, providing counterexamples and developing new methods for decomposition of the diagonal.

Contribution

It introduces two independent methods to prove decomposition of the diagonal in cases where the intermediate jacobian technique is not applicable.

Findings

Counterexample to stable rationality when X(R) is connected

Open question for irreducible geometric fibres

Development of new methods for decomposition of the diagonal

Abstract

We consider the question whether a real threefold X fibred into quadric surfaces over the real projective line is stably rational (over R) if the topological space X(R) is connected. We give a counterexample. When all geometric fibres are irreducible, the question is open. We investigate a family of such fibrations for which the intermediate jacobian technique is not available. We produce two independent methods which in many cases enable one to prove decomposition of the diagonal.