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Quotients of conic bundles. (English) Zbl 1375.14123

Let \(k\) be a field, \(\mathrm{char}\,k=0\). Let \(X\) be a \(k\)-rational surface \(X\) and \(G\) be its finite automorphism group. The main results of this paper are the following:
(1) The surface \(X/G\) is \(k\)-birationally equivalent to either a quotient of a \(k\)-rational del Pezzo surface by a finite automorphism group, or a quotient of a \(k\)-rational conic bundle by cyclic group of order \(2^m\), dyhedral group of order \(2^m\), alternating group of degree \(4\), symmetric group of degree \(4\), or alternating group of degree \(5\).
(2) Let \(|G|\) be odd, \(|G|> 10\), and \(|G|\neq 15\). If \(G\) is cyclic or \(|G|\) is not divisible by \(3\), then \(X/G\) is \(k\)-rational.

MSC:

14J26 Rational and ruled surfaces
14J50 Automorphisms of surfaces and higher-dimensional varieties
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