Belyi's theorem for smooth complete intersections of general type in generalized Grassmannians and weighted projective spaces

Cover Page

Cite item

Full Text

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

Abstract

We show that Javanpeykar's proof of Belyi's theorem for smooth complete intersections of general type in ordinary projective spaces can be generalised to smooth complete intersections of general type in generalised Grassmannians and weighted projective spaces. We propose an approach to the generalisation of this result to smooth complete intersections of general type in more general Mori dream spaces.

About the authors

Mikhail Aleksandrovich Ovcharenko

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia; International Laboratory for Mirror Symmetry and Automorphic Forms, National Research University Higher School of Economics, Moscow, Russia

Author for correspondence.
Email: ovcharenko@mi-ras.ru
without scientific degree, no status

References

  1. I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface, Cox rings, Cambridge Stud. Adv. Math., 144, Cambridge Univ. Press, Cambridge, 2015, viii+530 pp.
  2. I. Bauer, F. Catanese, F. Grunewald, “Faithful actions of the absolute Galois group on connected components of moduli spaces”, Invent. Math., 199:3 (2015), 859–888
  3. Г. В. Белый, “О расширениях Галуа максимального кругового поля”, Изв. АН СССР. Сер. матем., 43:2 (1979), 267–276
  4. W. Bruns, J. Herzog, Cohen–Macaulay rings, Cambridge Stud. Adv. Math., 39, 2nd rev. ed., Cambridge Univ. Press, Cambridge, 1998, xii+453 pp.
  5. A. Corti, M. Reid, “Weighted Grassmannians”, Algebraic geometry, A volume in memory of P. Francia, Walter de Gruyter & Co., Berlin, 2002, 141–163
  6. Groupes de monodromie en geometrie algebrique, Seminaire de geometrie algebrique du Bois-Marie 1967–1969 (SGA 7 II), v. II, Lecture Notes in Math., 340, eds. P. Deligne, N. Katz, Springer-Verlag, Berlin–New York, 1973, x+438 pp.
  7. A. Dimca, “Singularities and coverings of weighted complete intersections”, J. Reine Angew. Math., 1986:366 (1986), 184–193
  8. I. Dolgachev, “Weighted projective varieties”, Group actions and vector fields (Vancouver, BC, 1981), Lecture Notes in Math., 956, Springer-Verlag, Berlin, 1982, 34–71
  9. R. W. Easton, R. Vakil, “Absolute Galois acts faithfully on the components of the moduli space of surfaces: a Belyi-type theorem in higher dimension”, Int. Math. Res. Not. IMRN, 2007:20 (2007), rnm080, 10 pp.
  10. U. Garra, F. Zucconi, “Very ampleness and the infinitesimal Torelli problem”, Math. Z., 260:1 (2008), 31–46
  11. G. Gonzalez-Diez, “Belyi's theorem for complex surfaces”, Amer. J. Math., 130:1 (2008), 59–74
  12. A. Grothendieck, “Elements de geometrie algebrique. II. Etude globale elementaire de quelques classes de morphismes”, Inst. Hautes Etudes Sci. Publ. Math., 8 (1961), 5–222
  13. Р. Хартсхорн, Алгебраическая геометрия, Мир, М., 1981, 600 с.
  14. A. R. Iano-Fletcher, “Working with weighted complete intersections”, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000, 101–173
  15. A. Javanpeykar, “Belyi's theorem for complete intersections of general type”, Michigan Math. J., 66:1 (2017), 85–97
  16. S. J. Kovacs, M. Lieblich, “Erratum for {Boundedness} of families of canonically polarized manifolds: a higher dimensional analogue of Shafarevich's conjecture”, Ann. of Math. (2), 173:1 (2011), 585–617
  17. B. Köck, “Belyi's theorem revisited”, Beitr. Algebra Geom., 45:1 (2004), 253–265
  18. K. Konno, “Infinitesimal Torelli theorem for complete intersections in certain homogeneous Kähler manifolds. II”, Tohoku Math. J. (2), 42:3 (1990), 333–338
  19. Ph. Licht, “Infinitesimal Torelli for weighted complete intersections and certain Fano threefolds”, Beitr. Algebra Geom., 65:1 (2024), 97–127
  20. G. Malle, D. Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge Stud. Adv. Math., 133, Cambridge Univ. Press, Cambridge, 2011, xiv+309 pp.
  21. R. Muñoz, G. Occhetta, L. E. Sola Conde, K. Watanabe, J. A. Wisniewski, “A survey on the Campana–Peternell conjecture”, Rend. Istit. Mat. Univ. Trieste, 47 (2015), 127–185
  22. T. Okada, “Stable rationality of orbifold Fano threefold hypersurfaces”, J. Algebraic Geom., 28:1 (2019), 99–138
  23. M. Ovcharenko, “The classification of smooth well-formed Fano weighted complete intersections”, Internat. J. Math., 34:11 (2023), 2350064, 34 pp.
  24. M. Pieropan, “On Galois descent of complete intersections”, Math. Res. Lett., 28:4 (2021), 1243–1254
  25. V. V. Przyjalkowski, C. Shramov, Weighted complete intersections, De Gruyter Exp. Math., Walter de Gruyter GmbH, Berlin, 2024, x+250 pp.
  26. В. В. Пржиялковский, К. А. Шрамов, “Автоморфизмы взвешенных полных пересечений”, Алгебра, теория чисел и алгебраическая геометрия, Сборник статей. Посвящается памяти академика Игоря Ростиславовича Шафаревича, Труды МИАН, 307, МИАН, М., 2019, 217–229
  27. V. Przyjalkowski, C. Shramov, “Bounds for smooth Fano weighted complete intersections”, Commun. Number Theory Phys., 14:3 (2020), 511–553
  28. В. В. Пржиялковский, К. А. Шрамов, “Взвешенные полные пересечения Фано большой коразмерности”, Сиб. матем. журн., 61:2 (2020), 377–384
  29. В. В. Пржиялковский, К. А. Шрамов, “Об автоморфизмах квазигладких взвешенных полных пересечений”, Матем. сб., 212:3 (2021), 112–127
  30. M. Pizzato, T. Sano, L. Tasin, “Effective nonvanishing for Fano weighted complete intersections”, Algebra Number Theory, 11:10 (2017), 2369–2395
  31. T. Sano, L. Tasin, Delta invariants of weighted hypersurfaces
  32. T. Szamuely, Galois groups and fundamental groups, Cambridge Stud. Adv. Math., 117, Cambridge Univ. Press, Cambridge, 2009, x+270 pp.
  33. M. I. Qureshi, M. Wrobel, “Smooth Fano intrinsic Grassmannians of type $(2, n)$ with Picard number two”, Int. Math. Res. Not. IMRN, 2022:22 (2022), 17999–18034
  34. M. I. Qureshi, B. Szendrői, “Constructing projective varieties in weighted flag varieties”, Bull. Lond. Math. Soc., 43:4 (2011), 786–798
  35. M. I. Qureshi, “Constructing projective varieties in weighted flag varieties. II”, Math. Proc. Cambridge Philos. Soc., 158:2 (2015), 193–209
  36. M. I. Qureshi, B. Szendrői, “Calabi–Yau threefolds in weighted flag varieties”, Adv. High Energy Phys., 2012 (2012), 547317, 14 pp.
  37. S. Usui, “Local Torelli theorem for some non-singular weighted complete intersections”, Proceedings of the international symposium on algebraic geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store Co., Ltd., Tokyo, 1978, 723–734

Supplementary files

Supplementary Files
Action
1. JATS XML

Copyright (c) 2025 Ovcharenko M.A.

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).