Duke Mathematical Journal

Negative curves on algebraic surfaces

Thomas Bauer, Brian Harbourne, Andreas Leopold Knutsen, Alex Küronya, Stefan Müller-Stach, Xavier Roulleau, and Tomasz Szemberg

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study curves of negative self-intersection on algebraic surfaces. In contrast to what occurs in positive characteristics, it turns out that any smooth complex projective surface X with a surjective nonisomorphic endomorphism has bounded negativity (i.e., that C2 is bounded below for prime divisors C on X). We prove the same statement for Shimura curves on quaternionic Shimura surfaces of Hilbert modular type. As a byproduct, we obtain that there exist only finitely many smooth Shimura curves on such a surface. We also show that any set of curves of bounded genus on a smooth complex projective surface must have bounded negativity.

Article information

Duke Math. J., Volume 162, Number 10 (2013), 1877-1894.

First available in Project Euclid: 11 July 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
Secondary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]


Bauer, Thomas; Harbourne, Brian; Knutsen, Andreas Leopold; Küronya, Alex; Müller-Stach, Stefan; Roulleau, Xavier; Szemberg, Tomasz. Negative curves on algebraic surfaces. Duke Math. J. 162 (2013), no. 10, 1877--1894. doi:10.1215/00127094-2335368. https://projecteuclid.org/euclid.dmj/1373546606

Export citation


  • [1] W. L. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528.
  • [2] W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact Complex Surfaces, 2nd ed., Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin, 2004.
  • [3] W. Barth and I. Nieto, Abelian surfaces of type $(1,3)$ and quartic surfaces with $16$ skew lines, J. Algebraic Geometry 3 (1994), 173–222.
  • [4] G. Barthel, F. Hirzebruch, and T. Höfer, Geradenkonfigurationen und Algebraische Flächen, Aspects Math. D4, Vieweg & Sohn, Braunschweig, 1987.
  • [5] T. Bauer, Smooth Kummer surfaces in projective three-space, Proc. Amer. Math. Soc. 125 (1997), 2537–2541.
  • [6] T. Bauer, C. Bocci, S. Cooper, S. Di Rocco, M. Dumnicki, B. Harbourne, K. Jabbusch, A. L. Knutsen, A. Küronya, R. Miranda, J. Roé, H. Schenck, T. Szemberg, and Z. Teitler, “Recent developments and open problems in linear series” in Contributions to Algebraic Geometry, EMS Ser. Congr. Rep., European Math. Soc., Zürich, 2012, 93–140.
  • [7] T. Bauer and C. Schulz, Seshadri constants on the self-product of an elliptic curve, J. Algebra 320 (2008), 2981–3005.
  • [8] A. Beauville, Complex Algebraic Surfaces, 2nd ed., London Math. Soc. Stud. Texts 34, Cambridge Univ. Press, Cambridge, 1996.
  • [9] D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17–50.
  • [10] P. Deligne, “Travaux de Shimura” in Séminaire Bourbaki 1970/1971, no. 389, Lecture Notes in Math. 244, Springer, Berlin, 1971, 123–165.
  • [11] C. Fontanari, Towards bounded negativity of self-intersection on general blown-up projective planes, Comm. Algebra. 40 (2012), 1762–1765.
  • [12] Y. Fujimoto, Endomorphisms of smooth projective $3$-folds with non-negative Kodaira dimension, Publ. Res. Inst. Math. Sci. 38 (2002), 33–92.
  • [13] H. Granath, On quaternionic Shimura surfaces, Ph.D. dissertation, Chalmers University of Technology, Gothenburg, Sweden, 2002.
  • [14] B. Harbourne, Global aspects of the geometry of surfaces, Ann. Univ. Paedagog. Crac. Stud. Math. 11 (2010), 5–41.
  • [15] J. Harris, The interpolation problem, workshop lecture at “Classical Algebraic Geometry Today,” Mathematical Sciences Research Institute, Berkeley, California, 2009, http://jessica2.msri.org/attachments/13549/13549.pdf.
  • [16] R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • [17] A. Hurwitz, Über Riemannsche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 103 (1891), 1–60.
  • [18] R. Lazarsfeld, Positivity in Algebraic Geometry, I–II, Ergeb. Math. Grenzgeb. (3) 48–49; Springer, Berlin, 2004.
  • [19] S.-S. Lu and Y. Miyaoka, Bounding curves in algebraic surfaces by genus and Chern numbers, Math. Res. Lett. 2 (1995), 663–676.
  • [20] Y. Miyaoka, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), 159–171.
  • [21] Y. Miyaoka, The orbibundle Miyaoka-Yau-Sakai inequality and an effective Bogomolov-McQuillan theorem, Publ. Res. Inst. Math. Sci. 44 (2008), 403–417.
  • [22] S. Müller-Stach, E. Viehweg, and K. Zuo, Relative proportionality for subvarieties of moduli spaces of K3 and abelian surfaces, Pure Appl. Math. Q. 5 (2009), 1161–1199.
  • [23] D. Mumford, Hirzebruch’s proportionality theorem in the non-compact case, Invent. Math. 42 (1977), 239–272.
  • [24] N. Nakayama, Ruled surfaces with non-trivial surjective endomorphisms, Kyushu J. Math. 56 (2002), 433–446.
  • [25] I. H. Shavel, A class of algebraic surfaces of general type constructed from quaternion algebras, Pacific J. Math. 76 (1978), 221–245.