Rational points of rationally simply connected varieties
This is a survey of the author's work with [\textit{A. J. De Jong, X. He} and \textit{J. M. Starr}, Publ. Math., Inst. Hautes Étud. Sci. 114, 1--85 (2011; Zbl 1285.14053)] on Serre's conjecture II in the case of the function field \(K(S)\) of a surface \(S\) over an algebraically closed field \(k\): every \(G\) torsor over \(K(S)\) of a connected, simply connected, and semisimple algebraic group \(G\) over \(k\) is trivial, and more generally, a theorem about the existence of rational points for varieties defined over \(K(S)\).NEWLINENEWLINEThe proof uses a geometric argument involving rational simple connectedness. The general principle is that a variety over the field \(K(S)\) should have a rational point if it is ``rationally simply connected'' and if the elementary obstruction vanishes. However note that the definition of ``rational simple connectedness'' has been worked out only in the case of homogenous spaces ([Zbl 1285.14053] and [\textit{Y. Zhu}, ``Homogeneous fibrations over curves'', \url{arXiv:1111.2963}], and complete intersections [Zbl 1285.14053].NEWLINENEWLINERoughly speaking, rational simple connectedness is the analogue of simple connectedness in topology, just as rational connectedness can be thought of as the analogue of connectedness. So part of the definition involves defining what it means for the moduli space of rational curves to be rationally connected. The actual definition is too technical to state in a review.NEWLINENEWLINEThe idea of producing the rational point is the following (again this is an over-simplified account of the proof). One first finds a model of the variety over \(K(S)\) as an algebraic fibration over the surface \(S\). Then using the Lefschetz fibration, one can write this as a family of varieties over a curve defined over \(\kappa\), the function field of a curve (in this case, \(\mathbb{P}^1\)). It suffices to show that there is a section (over \(\kappa\)) of the family. To do this, one studies the space of sections and the Abel-Jacobi map to the Picard variety of the curve. If one can prove that this is surjective with rationally connected fibers, then one can conclude the existence of a section, which is the same as a rational point over \(\kappa\) of the space of sections, by a theorem of \textit{T. Graber, J. Harris} and \textit{J. M. Starr} [J. Am. Math. Soc. 16, No. 1, 57--67 (2003; Zbl 1092.14063)].NEWLINENEWLINEFor the entire collection see [Zbl 1237.14006].
scientific article; zbMATH DE number 6077521