Kunyavskiĭ, B. È.; Tsfasman, M. A. 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 Cited in 2 Reviews MSC: 14M20 Rational and unirational varieties 14J25 Special surfaces 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties Keywords:rationality of Néron-Severi torus; rational surface Citations:Zbl 0556.14027 × Cite Format Result Cite Review PDF Full Text: DOI