Stable birational invariants and the Lüroth problem

In dimension 1 and 2 and over an algebraically closed field of characteristic 0, unirationality implies rationality (hence a fortiori stable rationality). In the surface case, this follows from the Castenuovo characterization of smooth projective rational surfaces as those smooth projective surfaces X satisfying q(X) = p2(X) = 0 (see [5]). Starting from dimension 3, the answer to the Lüroth problem asking whether unirational varieties are rational is negative. Three methods have been developed in the 70’s to solve this problem, namely the Clemens-Griffiths method [11], the Iskovskikh-Manin method [23], and the Artin-Mumford method [2]. Among them, only the Artin-Mumford method solves the stable Lüroth problem, exhibiting unirational threefolds which are not stably rational. The invariant used by Artin-Mumford is the Brauer group and it is a topological invariant for rationally connected varieties. Colliot-Thélène and Ojanguren [12] described the higher degree generalization of this invariant, which takes the form of unramified cohomology with torsion coefficients. The first next invariant is the group H nr(X,Q/Z) which was reinterpreted in [14] as the group Hdg(X,Z)/H(X,Z)alg, at least when X is rationally connected smooth projective defined over C. It was proved in [40] that this group is trivial if X is rationally connected of dimension 3, and in fact the Colliot-Thélène-Ojanguren examples of unirational varieties for which this group is nontrivial are known to exist only starting from dimension 6 (it is likely that they exist starting from dimension 4).