Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quotients of del Pezzo surfaces of high degree

Author: Andrey Trepalin
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 14E08, 14M20; Secondary 14E07
Published electronically: December 27, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [AN06] Valery Alexeev and Viacheslav V. Nikulin, Del Pezzo and $ K3$ surfaces, MSJ Memoirs, vol. 15, Mathematical Society of Japan, Tokyo, 2006. MR 2227002
  • [Dol12] Igor V. Dolgachev, Classical algebraic geometry: A modern view, Cambridge University Press, Cambridge, 2012. MR 2964027
  • [DI09] 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, Inc., Boston, MA, 2009, pp. 443-548. MR 2641179,
  • [Isk79] 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
  • [Isk96] 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,
  • [Man67] Ju. I. Manin, Rational surfaces over perfect fields. II, Mat. Sb. (N.S.) 72 (114) (1967), 161-192 (Russian). MR 0225781
  • [Man74] Yu. I. Manin and M. Hazewinkel, Cubic forms: algebra, geometry, arithmetic, with translated from the Russian by M. Hazewinkel, North-Holland Mathematical Library, Vol. 4, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1974. MR 0460349
  • [Pop14] 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,
  • [Tr14] Andrey S. Trepalin, Rationality of the quotient of $ \mathbb{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,
  • [Tr16] Andrey Trepalin, Quotients of conic bundles, Transform. Groups 21 (2016), no. 1, 275-295. MR 3459712,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14E08, 14M20, 14E07

Retrieve articles in all journals with MSC (2010): 14E08, 14M20, 14E07

Additional Information

Andrey Trepalin
Affiliation: Institute for Information Transmission Problems, 19 Bolshoy Karetnyi side-street, Moscow 127994, Russia — and — Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 6 Usacheva street, Moscow 119048, Russia

Received by editor(s): June 2, 2015
Received by editor(s) in revised form: October 12, 2016
Published electronically: December 27, 2017
Additional Notes: The author was supported by the Russian Academic Excellence Project ‘5–100’, Young Russian Mathematics award, and the grant RFFI 15-01-02164-a
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society