Certaines fibrations en surfaces quadriques réelles

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.

💡 Research Summary

The paper investigates the rationality properties of real threefolds X that are fibred over the real projective line ℙ¹_R with fibres that are quadric surfaces. The central question is whether the topological condition “X(R) is connected” forces X to be stably rational over ℝ, or at least universally CH₀‑trivial (i.e. A₀(X_F)=0 for every field extension F/ℝ). The authors give a negative answer in general and develop new tools to treat the remaining cases where known techniques fail.

1. Counterexample when the Brauer group is non‑trivial.
The authors construct an explicit fibration X → ℙ¹_R whose generic fibre is a smooth quadric surface, but where some geometric fibres are reducible (so the fibration is not of type (I)). They show that X(R) is connected while Br(X)/Br(ℝ) ≠ 0. Since a stably rational variety over an infinite field must have Br(X)=Br(ℝ) (the Brauer group is a birational invariant for such varieties), this example proves that connectedness of the real locus does not imply stable rationality. Consequently X is also not universally CH₀‑trivial and not retract rational.

2. The remaining case: all geometric fibres irreducible (type (I)).
When every geometric fibre is integral, the Brauer group of X is trivial, so the Brauer‑obstruction disappears. The authors therefore turn to a higher unramified cohomology invariant. Let Δ be the discriminant curve (a smooth double cover of ℙ¹_R) associated with the fibration, and set W = X ×_R Δ. Inside the third Galois cohomology group H³(ℝ(W),ℤ/2) they consider the symbol \