zbMATH — the first resource for mathematics

Rational points on fibered surfaces. (English) Zbl 1101.14027
Tschinkel, Yuri (ed.), Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Summer Term 2004.; Lecture notes from the seminars “Number theory”, “Algebraic geometry” and “Geometric methods in representation theory” held at the University of Göttingen, Göttingen, Germany, 2004. Göttingen: Universitätsdrucke Göttingen (ISBN 3-930457-70-9/pbk). 103-109 (2004).
Rational surfaces, i.e. surfaces birational to \({\mathbb P}^2\) after an extension of the ground field, fall into two birational classes (not permitting ground field extension, and excluding \({\mathbb P}^2\) itself): conic bundles, birational to the surfaces \(a_0(U,V){X_0}^2+a_1(U,V)X_1^2+a_2(U,V)X_2^2=0\), where the \(a_i(U,V)\) are polynomials homogeneous of the same degree, and del Pezzo surfaces of degree \(d\), i.e. \({\mathbb P}^2\) blown up at 9-d points, where \(1 \leq d \leq 9\).
From the paper: “In both these cases the main conjecture is that the only obstruction to either the Hasse principle or weak approximation is the Brauer-Manin obstruction. But to work on these problems one hardly needs to know what the Brauer-Manin obstruction is. In my view, the right way to study the obstruction to the Hasse principle (for example) for any particular family of rational varieties is to find a sufficient condition for solubility everywhere locally to imply solubility globally, and only then to see how this condition compares with the Brauer-Manin condition” (weak approximation for a variety \(V\) means that \(V({\mathbb Q})\) is dense in \(\prod V({\mathbb Q}_v)\), where the product is over all primes of \({\mathbb Q}\)).
This philosophy is illustrated with studies of the arithmetic of conic bundles and of del Pezzo surfaces of degree 4.
For the entire collection see [Zbl 1082.11002].

14G05 Rational points