Certaines fibrations en surfaces quadriques réelles
Jean-Louis Colliot-Thélène, Alena Pirutka
TL;DR
The work tackles stable rationality of real threefolds X fibred by quadric surfaces over $P^1_{\mathbb R}$, producing a counterexample when $X(\mathbb R)$ is connected and developing two complementary methods to certify universal $CH_0$-triviality via nonramified cohomology and decomposition of the diagonal. Central to the approach is a discriminant double cover $\Delta$ and the cohomological invariant $(u+v,a,b)\in H^3_{nr}(k(X\times_k\Delta)/k,2)$, whose vanishing is shown to be equivalent to universal $CH_0$-triviality under precise hypotheses on the polynomial $p(u)$ and the base field. The paper develops both a general CTSk-based framework and a real-case analysis, including cases with and without complex multiplication on associated Jacobians, to obtain finite or vanishing obstruction groups and thus broad criteria for rationality, stable rationality, or non-rationality. These results illuminate the complex interplay between Brauer groups, discriminant curves, and Bloch–Ogus-type cohomology in real and arithmetic settings, and yield concrete families of both rational and non-rational quadric fibrations, with explicit examples and putative higher-dimensional generalizations. The findings advance understanding of when connected real loci fail to guarantee stable rationality and offer practical tools for testing universal CH$_0$-triviality across a range of quadric-fibration geometries.
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.