Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Multi-point Seshadri constants on ruled surfaces


Authors: Krishna Hanumanthu and Alapan Mukhopadhyay
Journal: Proc. Amer. Math. Soc. 145 (2017), 5145-5155
MSC (2010): Primary 14C20; Secondary 14H50
DOI: https://doi.org/10.1090/proc/13670
Published electronically: June 22, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a surface and let $ L$ be an ample line bundle on $ X$. We first obtain a lower bound for the Seshadri constant $ \varepsilon (X,L,r)$, when $ r \ge 2$. We then assume that $ X$ is a ruled surface and study Seshadri constants on $ X$ in greater detail. We also make precise computations of Seshadri constants on $ X$ in some cases.


References [Enhancements On Off] (What's this?)

  • [1] Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225
  • [2] F. Bastianelli, Remarks on the nef cone on symmetric products of curves, Manuscripta Math. 130 (2009), no. 1, 113-120. MR 2533770, https://doi.org/10.1007/s00229-009-0274-3
  • [3] Thomas Bauer, Seshadri constants on algebraic surfaces, Math. Ann. 313 (1999), no. 3, 547-583. MR 1678549, https://doi.org/10.1007/s002080050272
  • [4] Thomas Bauer, Sandra Di Rocco, Brian Harbourne, Michał Kapustka, Andreas Knutsen, Wioletta Syzdek, and Tomasz Szemberg, A primer on Seshadri constants, Interactions of classical and numerical algebraic geometry, Contemp. Math., vol. 496, Amer. Math. Soc., Providence, RI, 2009, pp. 33-70. MR 2555949, https://doi.org/10.1090/conm/496/09718
  • [5] Thomas Bauer and Tomasz Szemberg, Seshadri constants on surfaces of general type, Manuscripta Math. 126 (2008), no. 2, 167-175. MR 2403184, https://doi.org/10.1007/s00229-008-0170-2
  • [6] Arnaud Beauville, Complex algebraic surfaces, 2nd ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. MR 1406314
  • [7] Jean-Pierre Demailly, Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 87-104. MR 1178721, https://doi.org/10.1007/BFb0094512
  • [8] Lawrence Ein and Robert Lazarsfeld, Seshadri constants on smooth surfaces, Astérisque 218 (1993), 177-186. Journées de Géométrie Algébrique d'Orsay (Orsay, 1992). MR 1265313
  • [9] Łucja Farnik, A note on Seshadri constants of line bundles on hyperelliptic surfaces, Arch. Math. (Basel) 107 (2016), no. 3, 227-237. MR 3538518, https://doi.org/10.1007/s00013-016-0938-7
  • [10] Łucja Farnik, Tomasz Szemberg, Justyna Szpond and Halszka Tutaj-Gasińska, Restrictions on Seshadri constants on surfaces, arXiv:1602.08984v1, to appear in Taiwanese J. Math.
  • [11] Luis Fuentes García, Seshadri constants on ruled surfaces: the rational and the elliptic cases, Manuscripta Math. 119 (2006), no. 4, 483-505. MR 2223629, https://doi.org/10.1007/s00229-006-0629-y
  • [12] Krishna Hanumanthu, Positivity of line bundles on general blow ups of $ \mathbb{P}^2$, J. Algebra 461 (2016), 65-86. MR 3513065, https://doi.org/10.1016/j.jalgebra.2016.04.029
  • [13] Krishna Hanumanthu, Seshadri constants on surfaces with Picard number 1, arXiv:1608.04476, to appear in Manuscripta Math.
  • [14] Brian Harbourne, Seshadri constants and very ample divisors on algebraic surfaces, J. Reine Angew. Math. 559 (2003), 115-122. MR 1989646, https://doi.org/10.1515/crll.2003.044
  • [15] Brian Harbourne and Joaquim Roé, Discrete behavior of Seshadri constants on surfaces, J. Pure Appl. Algebra 212 (2008), no. 3, 616-627. MR 2365336, https://doi.org/10.1016/j.jpaa.2007.06.018
  • [16] Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR 0282977
  • [17] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • [18] Andreas Leopold Knutsen, Wioletta Syzdek, and Tomasz Szemberg, Moving curves and Seshadri constants, Math. Res. Lett. 16 (2009), no. 4, 711-719. MR 2525035, https://doi.org/10.4310/MRL.2009.v16.n4.a12
  • [19] Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472
  • [20] Keiji Oguiso, Seshadri constants in a family of surfaces, Math. Ann. 323 (2002), no. 4, 625-631. MR 1921551, https://doi.org/10.1007/s002080200317
  • [21] Syzdek, Wioletta, Seshadri constants and geometry of surfaces, Ph.D. thesis, 2005, University of Duisburg-Essen.
  • [22] Wioletta Syzdek and Tomasz Szemberg, Seshadri fibrations of algebraic surfaces, Math. Nachr. 283 (2010), no. 6, 902-908. MR 2668430, https://doi.org/10.1002/mana.200610121
  • [23] Szemberg Tomasz Global and local positivity of line bundles, Habilitation 2001.
  • [24] Tomasz Szemberg, Bounds on Seshadri constants on surfaces with Picard number 1, Comm. Algebra 40 (2012), no. 7, 2477-2484. MR 2948840, https://doi.org/10.1080/00927872.2011.579589
  • [25] Tomasz Szemberg, An effective and sharp lower bound on Seshadri constants on surfaces with Picard number 1, J. Algebra 319 (2008), no. 8, 3112-3119. MR 2408308, https://doi.org/10.1016/j.jalgebra.2007.10.036
  • [26] Geng Xu, Curves in $ {\bf P}^2$ and symplectic packings, Math. Ann. 299 (1994), no. 4, 609-613. MR 1286887, https://doi.org/10.1007/BF01459801

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14C20, 14H50

Retrieve articles in all journals with MSC (2010): 14C20, 14H50


Additional Information

Krishna Hanumanthu
Affiliation: Chennai Mathematical Institute, H1 SIPCOT IT Park, Siruseri, Kelambakkam 603103, India
Email: krishna@cmi.ac.in

Alapan Mukhopadhyay
Affiliation: Chennai Mathematical Institute, H1 SIPCOT IT Park, Siruseri, Kelambakkam 603103, India
Email: alapan@cmi.ac.in

DOI: https://doi.org/10.1090/proc/13670
Received by editor(s): October 28, 2016
Received by editor(s) in revised form: January 10, 2017
Published electronically: June 22, 2017
Additional Notes: The authors were partially supported by a grant from Infosys Foundation
Communicated by: Lev Borisov
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society