On automorphisms of quasi-smooth weighted complete intersections

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Abstract

We show that every reductive subgroup of the automorphism group of a quasi-smooth well-formed weighted complete intersection of dimension at least $3$ is a restriction of a subgroup in the automorphism group in the ambient weighted projective space. Also, we provide examples demonstrating that the automorphism group of a quasi-smooth well-formed Fano weighted complete intersection may be infinite and even non-reductive. Bibliography: 25 titles.

About the authors

Victor Vladimirovich Przyjalkowski

Steklov Mathematical Institute of Russian Academy of Sciences; International laboratory for Mirror Symmetry and Automorphic Forms, National Research University "Higher School of Economics" (HSE)

Email: victorprz@mi-ras.ru
Doctor of physico-mathematical sciences, no status

Constantin Aleksandrovich Shramov

Steklov Mathematical Institute of Russian Academy of Sciences; Laboratory of algebraic geometry and its applications, National Research University "Higher School of Economics" (HSE)

Email: costya.shramov@gmail.com
Doctor of physico-mathematical sciences, no status

References

  1. Э. Б. Винберг, А. Л. Онищик, Семинар по группам Ли и алгебраическим группам, 2-е изд., УРСС, М., 1995, 344 с.
  2. В. В. Пржиялковский, К. А. Шрамов, “Автоморфизмы взвешенных полных пересечений”, Алгебра, теория чисел и алгебраическая геометрия, Сборник статей. Посвящается памяти академика Игоря Ростиславовича Шафаревича, Тр. МИАН, 307, МИАН, М., 2019, 217–229
  3. В. В. Пржиялковский, К. А. Шрамов, “Взвешенные полные пересечения Фано большой коразмерности”, Сиб. матем. журн., 61:2 (2020), 377–384
  4. А. Н. Тюрин, “О пересечении квадрик”, УМН, 30:6(186) (1975), 51–99
  5. C. Ciliberto, R. Lazarsfeld, “On the uniqueness of certain linear series on some classes of curves”, Complete intersections (Acireale, 1983), Lecture Notes in Math., 1092, Springer, Berlin, 1984, 198–213
  6. A. Dimca, “Singularities and coverings of weighted complete intersections”, J. Reine Angew. Math., 1986:366 (1986), 184–193
  7. I. Dolgachev, “Weighted projective varieties”, Group actions and vector fields (Vancouver, BC, 1981), Lecture Notes in Math., 956, Berlin, Springer, 1982, 34–71
  8. D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Grad. Texts in Math., 150, Springer-Verlag, New York, 1995, xvi+785 pp.
  9. R. M. Fossum, The divisor class group of a Krull domain, Ergeb. Math. Grenzgeb., 74, Springer-Verlag, New York–Heidelberg, 1973, viii+148 pp.
  10. R. Hartshorne, “Complete intersections and connectedness”, Amer. J. Math., 84:3 (1962), 497–508
  11. R. Hartshorne, “Generalized divisors on Gorenstein curves and a theorem of Noether”, J. Math. Kyoto Univ., 26:3 (1986), 375–386
  12. 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
  13. Y. Kawamata, “The cone of curves of algebraic varieties”, Ann. of Math. (2), 119:3 (1984), 603–633
  14. A. R. Mavlyutov, “Cohomology of complete intersections in toric varieties”, Pacific J. Math., 191:1 (1999), 133–144
  15. Y. Miyaoka, S. Mori, “A numerical criterion for uniruledness”, Ann. of Math. (2), 124:1 (1986), 65–69
  16. T. Okada, “Stable rationality of orbifold Fano 3-fold hypersurfaces”, J. Algebraic Geom., 28:1 (2019), 99–138
  17. M. Pizzato, T. Sano, L. Tasin, “Effective nonvanishing for Fano weighted complete intersections”, Algebra Number Theory, 11:10 (2017), 2369–2395
  18. Yu. Prokhorov, C. Shramov, “Jordan constant for Cremona group of rank 3”, Mosc. Math. J., 17:3 (2017), 457–509
  19. V. Przyjalkowski, C. Shramov, “Bounds for smooth Fano weighted complete intersections”, Commun. Number Theory Phys., 14:3 (2020), 511–553
  20. V. Przyjalkowski, C. Shramov, “Hodge level for weighted complete intersections”, Collect. Math., 71:3 (2020), 549–574
  21. L. Robbiano, “Some properties of complete intersections in “good” projective varieties”, Nagoya Math. J., 61 (1976), 103–111
  22. M. Rossi, L. Terracini, “Linear algebra and toric data of weighted projective spaces”, Rend. Semin. Mat. Univ. Politec. Torino, 70:4 (2012), 469–495
  23. K. Ueno, Classification theory of algebraic varieties and compact complex spaces, Notes written in collaboration with P. Cherenack, Lecture Notes in Math., 439, Springer-Verlag, Berlin–New York, 1975, xix+278 pp.
  24. K. Watanabe, “Some remarks concerning Demazure's construction of normal graded rings”, Nagoya Math. J., 83 (1981), 203–211
  25. Qi Zhang, “Rational connectedness of $log Q$-Fano varieties”, J. Reine Angew. Math., 2006:590 (2006), 131–142

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