Skip to main content
Log in

Automorphisms of the Hilbert schemes of n points of a rational surface and the anticanonical Iitaka dimension

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

The purpose of this paper is to investigate relationship between the automorphism group of a rational surface and that of its Hilbert scheme of n points.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Arapura, D., Archava, S.: Kodaira dimension of symmetric powers. Proc. Am. Math. Soc. 131(5), 1369–1372 (2003). (electronic)

    Article  MathSciNet  Google Scholar 

  2. Arcara, D., Bertram, A., Coskun, I., Huizenga, J.: The minimal model program for the Hilbert scheme of points on \({\mathbb{P}}^{2}\) and Bridgeland stability. Adv. Math. 235, 580–626 (2013). (English summary)

    Article  MathSciNet  Google Scholar 

  3. Artebani, M., Laface, A.: Cox rings of surfaces and the anticanonical Iitaka dimension. Adv. Math. 226(6), 5252–5267 (2011)

    Article  MathSciNet  Google Scholar 

  4. Beauville, A.: Some remarks on Kähler manifolds with c\(_{1}\)=0. In: Ueno, K. (ed.) Classification of Algebraic and Analytic Manifolds (Katata, 1982), pp. 1–26. Progr. Math., vol. 39. Birkhäuser, Boston (1983)

  5. Belmans, P., Oberdieck, G., Rennemo, J.: Automorphisms of Hilbert Schemes of Points on Surfaces. arXiv:1907.07064

  6. Bertram, A., Coskun, I.: The birational geometry of the Hilbert schemes of points on surfaces. In: Bogomolov, F., Hassett, B., Tschinkel, Y. (eds.) Birational Geometry, Rational Curves, and Arithmetic, pp. 15–55. Springer, New York (2013)

    Chapter  Google Scholar 

  7. Boissière, S.: Automorphismes naturels de l’espace de Douady de points sur une surface. Can. J. Math. 64(1), 3–23 (2012)

    Article  MathSciNet  Google Scholar 

  8. Boissière, S., Sarti, A.: A note on automorphisms and birational transformations of holomorphic symplectic manifolds. Proc. Am. Math. Soc. 140(12), 4053–4062 (2012)

    Article  MathSciNet  Google Scholar 

  9. Boissière, S., Cattaneo, A., Nieper-Wisskirchen, M., Sarti, A.: Higher dimensional enriques varieties and automorphisms of generalized Kummer varieties. J. Math. Pures Appl. (9) 95(5), 553–563 (2011)

    Article  MathSciNet  Google Scholar 

  10. Boissière, S., Cattaneo, A., Nieper-Wisskirchen, M., Sarti, A.: The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface. In: Proceedings of the Schiermonnikoog conference, K3 surfaces and their moduli, Progress in Mathematics 315. Birkhäuser (2015)

  11. Bolognese, B., Huizenga, J., Lin, Y., Riedl, E., Schmidt, B., Woolf, M., Zhao, X.: Nef cones of Hilbert schemes of points on surfaces. Algebra Number Theory 10(4), 907–930 (2016)

    Article  MathSciNet  Google Scholar 

  12. Brion, M., Kumar, S.: Frobenius splitting methods in geometry and representation theory. Progress in Mathematics, vol. 231. Birkhäuser, Boston (2005)

    Book  Google Scholar 

  13. Cattaneo, A.: Automorphisms of Hilbert Schemes of Points on a Generic Projective K3 Surface. arXiv:1801.05682

  14. Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math. 90, 511–521 (1968)

    Article  MathSciNet  Google Scholar 

  15. Fogarty, J.: Algebraic families on an algebraic surface II: The Picard scheme of the punctual Hilbert scheme. Am. J. Math. 95, 660–687 (1973)

    Article  MathSciNet  Google Scholar 

  16. Göttsche, L., Soergel, W.: Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces. Math. Ann. 296, 235–245 (1993)

    Article  MathSciNet  Google Scholar 

  17. Hassett, B., Tschinkel, Y.: Extremal Rays and Automorphisms of Holomorphic Symplectic Varieties. arXiv:1506.08153

  18. Hayashi, T.: Universal covering calabi-yau manifolds of the hilbert schemes of n points of enriques surfaces. Asian J. Math 21(6), 1099–1120 (2017)

    Article  MathSciNet  Google Scholar 

  19. Huybrechts, D., Lehnl, M.: The Geometry of Moduli Spaces of Sheaves. Aspects of Mathematics E31. Friedr. Vieweg and Sohn, Braunschweig (1997)

    Book  Google Scholar 

  20. Li, W.-P., Qin, Z., Zhang, Q.: Curves in the Hilbert schemes of points on surfaces. Contemp. Math. 322, 89–96 (2003)

    Article  MathSciNet  Google Scholar 

  21. Oguiso, K.: On automorphisms of the punctual Hilbert schemes of K3 surfaces. Eur. J. Math. 2(1), 246–261 (2016)

    Article  MathSciNet  Google Scholar 

  22. Sosna, P.: Fourier-Mukai partners of canonical covers of bielliptic and enriques surfaces. Rend. Sem. Mat. Univ. Padova 130, 203–213 (2013)

    Article  MathSciNet  Google Scholar 

  23. Testa, D., Várilly-Alvarado, A., Velasco, M.: Big rational surfaces. Math. Ann. 351(1), 95–107 (2011)

    Article  MathSciNet  Google Scholar 

  24. Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321(4), 817–884 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Taro Hayashi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hayashi, T. Automorphisms of the Hilbert schemes of n points of a rational surface and the anticanonical Iitaka dimension. Geom Dedicata 207, 395–407 (2020). https://doi.org/10.1007/s10711-019-00504-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-019-00504-7

Keywords

Mathematics Subject Classification (2010)

Navigation