Forms of del Pezzo surfaces of degree 5 and 6

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

We obtain necessary and sufficient condition for the existence of del Pezzo surfaces of degrees 5">5 and 6">6 over a field K">K with a prescribed action of absolute Galois group Gal⁡(Ksep/K)">Gal⁡(Ksep/K) on the graph of (−1)">(−1)-curves. We also compute the automorphism groups of del Pezzo surfaces of degree 5">5 over arbitrary fields.

About the authors

Alexandr Vladimirovich Zaitsev

Laboratory of algebraic geometry and its applications, National Research University "Higher School of Economics" (HSE)

Author for correspondence.
Email: math-net2025_06@mi-ras.ru

without scientific degree, no status

References

  1. I. V. Dolgachev, V. A. Iskovskikh, “Finite subgroups of the plane Cremona group”, Algebra, arithmetic, and geometry, In honor of Yu. I. Manin, v. I, Progr. Math., 269, Birkhäuser Boston, Boston, MA, 2009, 443–548
  2. I. V. Dolgachev, V. A. Iskovskikh, “On elements of prime order in the plane Cremona group over a perfect field”, Int. Math. Res. Not. IMRN, 2009:18 (2009), 3467–3485
  3. E. A. Yasinsky, Automorphisms of real del Pezzo surfaces and the real plane Cremona group
  4. J.-P. Serre, “A Minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field”, Mosc. Math. J., 9:1 (2009), 183–198
  5. A. Beauville, “Finite subgroups of $mathrm{PGL}_2(K)$”, Vector bundles and complex geometry, Contemp. Math., 522, Amer. Math. Soc., Providence, RI, 2010, 23–29
  6. M. Garcia-Armas, “Finite group actions on curves of genus zero”, J. Algebra, 394 (2013), 173–181
  7. I. V. Dolgachev, Classical algebraic geometry. A modern view, Cambridge Univ. Press, Cambridge, 2012, xii+639 pp.
  8. A. Skorobogatov, Torsors and rational points, Cambridge Tracts in Math., 144, Cambridge Univ. Press, Cambridge, 2001, viii+187 pp.
  9. Hsueh-Yung Lin, E. Shinder, S. Zimmermann, Factorization centers in dimension two and the Grothendieck ring of varieties
  10. И. Р. Шафаревич, Основы алгебраической геометрии, 3-е изд., МЦНМО, М., 2007, 590 с.
  11. The Stacks project
  12. A. N. Skorobogatov, “On a theorem of Enriques–Swinnerton-Dyer”, Ann. Fac. Sci. Toulouse Math. (6), 2:3 (1993), 429–440
  13. A. Auel, M. Bernardara, “Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields”, Proc. Lond. Math. Soc. (3), 117:1 (2018), 1–64
  14. Incomplete failures of the inverse Galois problem
  15. Б. Л. ван дер Варден, Алгебра, Наука, М., 1976, 648 с.
  16. И. Р. Шафаревич, “Построение полей алгебраических чисел с заданной разрешимой группой Галуа”, Изв. АН СССР. Сер. матем., 18:6 (1954), 525–578
  17. N. Vila, “On the inverse problem of Galois theory”, Publ. Mat., 36:2B (1992), 1053–1073
  18. C. O. Горчинский, К. А. Шрамов, Неразветвленная группа Брауэра и ее приложения, МЦНМО, М., 2018, 200 с.
  19. J. Schneider, S. Zimmermann, “Algebraic subgroups of the plane Cremona group over a perfect field”, Epijournal Geom. Algebrique, 5 (2021), 14, 48 pp.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2023 Zaitsev A.V.

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).