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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].

##### MSC:
 14G05 Rational points