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Automorphisms of Weighted Complete Intersections

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Abstract

We show that smooth well-formed weighted complete intersections have finite automorphism groups, with several obvious exceptions.

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References

  1. C. Araujo, M. Corrêa, and A. Massarenti, “Codimension one Fano distributions on Fano manifolds,” Commun. Contemp. Math. 20 (5), 1750058 (2018).

    Article  MathSciNet  Google Scholar 

  2. O. Benoist, “Séparation et propriété de Deligne-Mumford des champs de modules d’intersections complètes lisses,” J. London Math. Soc., Ser. 2, 87 (1), 138–156 (2013).

    Article  Google Scholar 

  3. A. Borel, Linear Algebraic Groups (W. A. Benjamin, New York, 1969).

    MATH  Google Scholar 

  4. I. Cheltsov and C. Shramov, Cremona Groups and the Icosahedron (CRC Press, Boca Raton, FL, 2016), Monogr. Res. Notes Math.

    MATH  Google Scholar 

  5. J.-J. Chen, J. A. Chen, and M. Chen, “On quasismooth weighted complete intersections,” J. Algebr. Geom. 20 (2), 239–262 (2011).

    Article  MathSciNet  Google Scholar 

  6. I. Dolgachev, “Weighted projective varieties,” in Group Actions and Vector Fields: Proc. Pol.-North Am. Semin., Vancouver, 1981 (Springer, Berlin, 1982), Lect. Notes Math. 956, pp. 34–71.

    Chapter  Google Scholar 

  7. I. V. Dolgachev, Classical Algebraic Geometry: A Modern View (Cambridge Univ. Press, Cambridge, 2012).

    Book  Google Scholar 

  8. H. Flenner, “Divisorenklassengruppen quasihomogener Singularitäten,” J. Reine Angew. Math. 328, 128–160 (1981).

    MathSciNet  MATH  Google Scholar 

  9. T. Fujita, “On the structure of polarized manifolds with total deficiency one. I,” J. Math. Soc. Japan 32, 709–725 (1980).

    Article  MathSciNet  Google Scholar 

  10. T. Fujita, “On the structure of polarized manifolds with total deficiency one. II,” J. Math. Soc. Japan 33, 415–434 (1981).

    Article  MathSciNet  Google Scholar 

  11. T. Fujita, “On the structure of polarized manifolds with total deficiency one. III,” J. Math. Soc. Japan 36, 75–89 (1984).

    Article  MathSciNet  Google Scholar 

  12. C. D. Hacon, J. McKernan, and C. Xu, “On the birational automorphisms of varieties of general type,” Ann. Math., Ser. 2, 177 (3), 1077–1111 (2013).

    Article  MathSciNet  Google Scholar 

  13. R. Hartshorne, Local Cohomology: A Seminar Given by A. Grothendieck, Harvard Univ., Fall, 1961 (Springer, Berlin, 1967), Lect. Notes Math. 41.

    Book  Google Scholar 

  14. R. Hartshorne, Algebraic Geometry (Springer, New York, 1977), Grad. Texts Math. 52.

    Book  Google Scholar 

  15. A. R. Iano-Fletcher, “Working with weighted complete intersections,” in Explicit Birational Geometry of 3-Folds (Cambridge Univ. Press, Cambridge, 2000), LMS Lect. Note Ser. 281, pp. 101–173.

    Chapter  Google Scholar 

  16. V. A. Iskovskikh and Yu. G. Prokhorov, Fano Varieties (Springer, Berlin, 1999), Encycl. Math. Sci. 47.

    MATH  Google Scholar 

  17. K. Kodaira and D. C. Spencer, “On deformations of complex analytic structures. II,” Ann. Math., Ser. 2, 67 (3), 403–466 (1958).

    Article  MathSciNet  Google Scholar 

  18. A. G. Kuznetsov, Yu. G. Prokhorov, and C. A. Shramov, “Hilbert schemes of lines and conics and automorphism groups of Fano threefolds,” Jpn. J. Math. 13 (1), 109–185 (2018).

    Article  MathSciNet  Google Scholar 

  19. A. R. Mavlyutov, “Cohomology of complete intersections in toric varieties,” Pac. J. Math. 191 (1), 133–144 (1999).

    Article  MathSciNet  Google Scholar 

  20. H. Matsumura and P. Monsky, “On the automorphisms of hypersurfaces,” J. Math. Kyoto Univ. 3, 347–361 (1964).

    Article  MathSciNet  Google Scholar 

  21. Y. Miyaoka and S. Mori, “A numerical criterion for uniruledness,” Ann. Math., Ser. 2, 124 (1), 65–69 (1986).

    Article  MathSciNet  Google Scholar 

  22. S. Mori, “On a generalization of complete intersections,” J. Math. Kyoto Univ. 15 (3), 619–646 (1975).

    Article  MathSciNet  Google Scholar 

  23. S. Mukai, “Curves, K3 surfaces and Fano 3-folds of genus ≤ 10,” in Algebraic Geometry and Commutative Algebra: In Honor of M. Nagata (Konokuniya, Tokyo, 1988), Vol. I, pp. 357–377.

    Chapter  Google Scholar 

  24. T. Okada, “Stable rationality of orbifold Fano 3-fold hypersurfaces,” J. Algebr. Geom. 28 (1), 99–138 (2019).

    Article  MathSciNet  Google Scholar 

  25. M. Pizzato, T. Sano, and L. Tasin, “Effective nonvanishing for Fano weighted complete intersections,” Algebra Number Theory 11 (10), 2369–2395 (2017).

    Article  MathSciNet  Google Scholar 

  26. V. V. Przyjalkowski, I. A. Cheltsov, and C. A. Shramov, “Fano threefolds with infinite automorphism groups,” Izv. Math. 83 (4), 860–907 (2019) [transl. from Izv. Ross. Akad. Nauk, Ser. Mat. 83 (4), 226–280 (2019)].

    Article  MathSciNet  Google Scholar 

  27. V. Przyjalkowski and C. Shramov, “Bounds for smooth Fano weighted complete intersections,” arXiv: 1611.09556 [math.AG].

  28. V. Przyjalkowski and C. Shramov, “Hodge level for weighted complete intersections,” Collect. Math., doi: https://doi.org/10.1007/s13348-019-00276-z (2020); arXiv: 1801.10489v2 [math.AG].

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Acknowledgments

We are grateful to B. Fu, B. van Geemin, A. Kuznetsov, A. Massarenti, and T. Sano for useful discussions. We also thank the referee for his suggestions regarding the first draft of the paper.

Funding

Victor Przyjalkowski was supported by the HSE Laboratory for Mirror Symmetry and Automorphic Forms (Russian Federation Government grant, contract no. 14.641.31.0001). Constantin Shramov was supported by the HSE Basic Research Program, the Russian Academic Excellence Project “5-100,” and the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS.” Both authors are Young Russian Mathematics award winners and would like to thank its sponsors and jury.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Victor V. Przyjalkowski & Constantin A. Shramov

  2. International Laboratory for Mirror Symmetry and Automorphic Forms, HSE University, ul. Usacheva 6, Moscow, 119048, Russia

    Victor V. Przyjalkowski

  3. Laboratory of Algebraic Geometry and its Applications, HSE University, ul. Usacheva 6, Moscow, 119048, Russia

    Constantin A. Shramov

Authors
  1. Victor V. Przyjalkowski
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  2. Constantin A. Shramov
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Corresponding authors

Correspondence to Victor V. Przyjalkowski or Constantin A. Shramov.

Additional information

This article was submitted by the authors simultaneously in Russian and English

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 307, pp. 217–229.

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Przyjalkowski, V.V., Shramov, C.A. Automorphisms of Weighted Complete Intersections. Proc. Steklov Inst. Math. 307, 198–209 (2019). https://doi.org/10.1134/S0081543819060129

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  • DOI: https://doi.org/10.1134/S0081543819060129

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