Quotients of del Pezzo surfaces of high degree

Trepalin, Andrey

doi:10.1090/tran/7130

Quotients of del Pezzo surfaces of high degree
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Trans. Amer. Math. Soc. 370 (2018), 6097-6124 Request permission

Abstract:

In this paper we study quotients of del Pezzo surfaces of degree four and more over arbitrary field $\Bbbk$ of characteristic zero by finite groups of automorphisms. We show that if a del Pezzo surface $X$ contains a point defined over the ground field and the degree of $X$ is at least five, then the quotient is always $\Bbbk$-rational. If the degree of $X$ is equal to four, then the quotient can be non-$\Bbbk$-rational only if the order of the group is $1$, $2$, or $4$. For these groups we construct examples of non-$\Bbbk$-rational quotients.

References

Valery Alexeev and Viacheslav V. Nikulin, Del Pezzo and $K3$ surfaces, MSJ Memoirs, vol. 15, Mathematical Society of Japan, Tokyo, 2006. MR 2227002, DOI 10.1142/e002
Igor V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012. A modern view. MR 2964027, DOI 10.1017/CBO9781139084437
Igor V. Dolgachev and Vasily A. Iskovskikh, Finite subgroups of the plane Cremona group, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser Boston, Boston, MA, 2009, pp. 443–548. MR 2641179, DOI 10.1007/978-0-8176-4745-2_{1}1
V. A. Iskovskih, Minimal models of rational surfaces over arbitrary fields, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 19–43, 237 (Russian). MR 525940
V. A. Iskovskikh, Factorization of birational mappings of rational surfaces from the point of view of Mori theory, Uspekhi Mat. Nauk 51 (1996), no. 4(310), 3–72 (Russian); English transl., Russian Math. Surveys 51 (1996), no. 4, 585–652. MR 1422227, DOI 10.1070/RM1996v051n04ABEH002962
Ju. I. Manin, Rational surfaces over perfect fields. II, Mat. Sb. (N.S.) 72 (114) (1967), 161–192 (Russian). MR 0225781
Yu. I. Manin and M. Hazewinkel, Cubic forms: algebra, geometry, arithmetic, North-Holland Mathematical Library, Vol. 4, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1974. Translated from the Russian by M. Hazewinkel. MR 0460349
Vladimir L. Popov, Jordan groups and automorphism groups of algebraic varieties, Automorphisms in birational and affine geometry, Springer Proc. Math. Stat., vol. 79, Springer, Cham, 2014, pp. 185–213. MR 3229352, DOI 10.1007/978-3-319-05681-4_{1}1
Andrey S. Trepalin, Rationality of the quotient of $\Bbb {P}^2$ by finite group of automorphisms over arbitrary field of characteristic zero, Cent. Eur. J. Math. 12 (2014), no. 2, 229–239. MR 3130680, DOI 10.2478/s11533-013-0340-7
Andrey Trepalin, Quotients of conic bundles, Transform. Groups 21 (2016), no. 1, 275–295. MR 3459712, DOI 10.1007/s00031-015-9342-9