##
**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.

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.

Reviewer: Nikolaos Tziolas (Nicosia)

### 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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\textit{A. Trepalin}, Trans. Am. Math. Soc. 370, No. 9, 6097--6124 (2018; Zbl 1406.14010)

### References:

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[7] | Manin, Yu. I.; Hazewinkel, M., Cubic forms: algebra, geometry, arithmetic, vii+292 pp. (1974), North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York |

[8] | Popov, Vladimir L., Jordan groups and automorphism groups of algebraic varieties. Automorphisms in birational and affine geometry, Springer Proc. Math. Stat. 79, 185-213 (2014), Springer, Cham · Zbl 1325.14024 |

[9] | Trepalin, Andrey S., Rationality of the quotient of \(\mathbb{P}^2\) by finite group of automorphisms over arbitrary field of characteristic zero, Cent. Eur. J. Math., 12, 2, 229-239 (2014) · Zbl 1288.14009 |

[10] | Trepalin, Andrey, Quotients of conic bundles, Transform. Groups, 21, 1, 275-295 (2016) · Zbl 1375.14123 |

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