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On the Néron-Severi torus of a rational surface. (English. Russian original) Zbl 0556.14026

Russ. Math. Surv. 37, No. 4, 121-122 (1982); translation from Usp. Mat. Nauk 37, No. 4(226), 163-164 (1982).
Let \(X\) be a complete geometrically irreducible rational surface over the algebraic number field \(k\), \(d=K_ X^ 2\) denote its degree and \(S\) the corresponding Néron-Severi torus. The first result is that \(S\) is rational if \(d\geq 5\) and that for \(d\leq 4\) one can find surfaces \(X\) as above with \(S\) nonrational.
Next, let \(L\) denote the decomposition field of \(X\) and \(G=\mathrm{Gal}(L/k)\). Then \(G\) is a subgroup in some Weyl group \(W\). In the case of Del Pezzo surfaces with \(d=4\) it is shown that the conjugacy class of \(G\) in \(W\) determines whether or not the torus \(S\) is rational. Some explicit examples are also constructed.
Reviewer: Alexandru Dimca

MSC:

14M20 Rational and unirational varieties
14J25 Special surfaces
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties

Citations:

Zbl 0556.14027
Full Text: DOI