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Quotients of del Pezzo surfaces of high degree. (English) Zbl 1406.14010

Let \(X\) be a variety defined over a field \(k\). \(X\) is called \(k\)-rational if and only if it is birational to \(\mathbb{P}^n_k\), for some \(n>0\). \(X\) is called geometrically rational if and only if \(\bar{X}=X\otimes_k \bar{k}\) is \(\bar{k}\)-rational, where \(\bar{k}\) is the algebraic closure of \(k\),
This paper studies the problem of when the quotient of a \(k\)-rational surface by a finite group is also \(k\)-rational. In general this is not true. In fact, the set of non \(k\)-rational quotients of rational surfaces is birationally unbounded [the author, Transform. Groups 21, No. 1, 275–295 (2016; Zbl 1375.14123)].
The main result of this paper is the following.
Let \(X\) be a del Pezzo surface defined over a field \(k\) of characteristic zero and such that \(X\) has a \(k\)-point (by a theorem of Manin \(X\) is \(k\)-rational). Let \(G\) be a finite group of automorphisms of \(X\). Suppose that \(K_X^2\geq 5\). Then the quotient surface \(X/G\) is also \(k\)-rational. Suppose that \(K_X^2=4\), the order of \(G\) is equal to 1,2 or 4 and all non-trivial elements of \(G\) have only isolated fixed points. Then \(X/G\) is not \(k\)-rational. In all other cases, \(X/G\) is \(k\)-rational.
This result combined with previous results of the author implies the following. Let \(X\) be a smooth rational surface defined over an algebraically closed field \(k\) such that \(X\) has a \(k\)-point. Let \(G\) be a finite group of automorphisms of \(X\). Then if \(K_X^2 \geq 5\), the quotient \(X/G\) is \(k\)-rational.

MSC:

14E08 Rationality questions in algebraic geometry
14M20 Rational and unirational varieties
14E07 Birational automorphisms, Cremona group and generalizations

Citations:

Zbl 1375.14123
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References:

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