## Duke Mathematical Journal

### Negative curves on algebraic surfaces

#### Abstract

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 $C^{2}$ 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

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

Dates
First available in Project Euclid: 11 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1373546606

Digital Object Identifier
doi:10.1215/00127094-2335368

Mathematical Reviews number (MathSciNet)
MR3079262

Zentralblatt MATH identifier
1272.14009

#### Citation

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

#### References

• [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.