Zero-cycles and rational points on some surfaces over a global function field (Q2919667)

The main object of the paper under review is a surface \(X\subset\mathbb P^ 3\), defined over \(k=\mathbb F(t)\), where \(\mathbb F\) is a finite field, by an equation of the form \(f+tg=0\), where \(f\) and \(g\) are nonproportional forms in 4 variables with coefficients in \(\mathbb F\) of degree \(d\) prime to the characteristic of \(\mathbb F\). The surface \(X\) is assumed smooth. The authors prove that for such an \(X\), the Brauer--Manin obstruction to the Hasse principle for zero-cycles of degree 1 is the only one. For \(d=3\), they prove that the Brauer--Manin obstruction to the Hasse principle for rational points is the only one. An important ingredient in the proofs of these results is a strong integral form of the Tate conjecture on 1-dimensional cycles (see \textit{S.~Saito} [Invent. Math. 98, No. 2, 371--404 (1989; Zbl 0694.14005)] or \textit{J.-L.~Colliot-Thélène} [Proc. Symp. Pure Math. 67, 1--12 (1999; Zbl 0981.14003)]). The last remark of the paper under review contains a sketch of an alternative proof of the second result, based on the approach taken in \textit{P.~Swinnerton--Dyer} [Math. Proc. Camb. Philos. Soc. 113, No. 3, 449--460 (1993; Zbl 0804.14018)], which could allow one to avoid the cohomological machinery in this case.