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Del Pezzo surfaces of degree four. (English) Zbl 0705.14039
The authors undertake an extensive study of Del Pezzo surfaces X of degree four, i.e. complete intersections of two quadrics in \({\mathbb{P}}^ 4_ K\). The authors’ principal goal is to investigate the interrelations between the arithmetical, algebraic and geometric properties of these rational surfaces. To this end, the authors study the action of the Galois group \(Gal(\bar k/k)\) on \(Pic(\bar X)\), where \(\bar k\) is the separable closure of k and \(\bar X=X_{Spec(k)}\times Spec(\bar k)\). The techniques used in the paper include Weyl groups and root systems, Galois cohomology, algebraic tori in semi-simple groups and conic bundles. The paper contains a lot of explicit examples.
Reviewer: F.L.Zak

MSC:
14J26 Rational and ruled surfaces
12G05 Galois cohomology
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