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Exceptional curves on smooth rational surfaces with $-K$ not nef and of self-intersection zero


Author: Mustapha Lahyane
Journal: Proc. Amer. Math. Soc. 133 (2005), 1593-1599
MSC (2000): Primary 14J26; Secondary 14F05
DOI: https://doi.org/10.1090/S0002-9939-04-07693-2
Published electronically: December 31, 2004
MathSciNet review: 2120267
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Abstract: A $(-n)$-curve is a smooth rational curve of self-intersection $-n$, where $n$ is a positive integer. In 1998 Hirschowitz asked whether a smooth rational surface $X$ defined over the field of complex numbers, having an anti-canonical divisor not nef and of self-intersection zero, has $(-2)$-curves. In this paper we prove that for such a surface $X$, the set of $(-1)$-curves on $X$ is finite but non-empty, and that $X$ may have no $(-2)$-curves. Related facts are also considered.


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

  • 1. W. Barth, C. Peters , A. Van de Ven. Compact Complex Surfaces. Berlin, Springer (1984). MR 0749574 (86c:32026)
  • 2. B. Harbourne, Anticanonical rational surfaces, Transactions of the American Mathematical Society, Volume 349 (1997), Number 3, 1191-1208. MR 1373636 (97f:14007)
  • 3. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer Verlag (1977). MR 0463157 (57:3116)
  • 4. M. Lahyane, Rational Surfaces Having Only a Finite Number of Exceptional Curves. Preprint of The Abdus Salam International Centre for Theoretical Physics, October 2001. Mathematische Zeitschrift, volume 247 (2004), 213-221. MR 2054527
  • 5. J. Rosoff, Effective divisor classes and blowings-up of $\mathbb{P}^{2}$. Pacific Journal of Mathematics Vol. 89, No.2, 419-429, 1980. MR 0599129 (82e:14042)
  • 6. R. Miranda, U. Persson, On Extremal Rational Elliptic Surfaces. Mathematische Zeitschrift 193, 537-558 (1986). MR 0867347 (88a:14044)
  • 7. M. Nagata, On rational surfaces, II, Mem. Coll. Sci. Univ. Kyoto, Ser.E A Math. 33 (1960), 271-293. MR 0126444 (23:A3740)

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Additional Information

Mustapha Lahyane
Affiliation: Abdus Salam International Centre for Theoretical Physics, 34100 Trieste, Italy
Address at time of publication: Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Vallodolid University, 47005 Valladolid, Spain
Email: lahyane@agt.uva.es

DOI: https://doi.org/10.1090/S0002-9939-04-07693-2
Keywords: Anticanonical rational surfaces, minimal models of smooth rational surfaces, Hodge Index Theorem, points in general position, N\'eron-Severi group, blowing-up
Received by editor(s): August 27, 2001
Received by editor(s) in revised form: February 23, 2004
Published electronically: December 31, 2004
Additional Notes: This work was partially supported by a postdoctoral fellowship at the International Centre for Theoretical Physics (Trieste, Italy) and by a Marie Curie grant number HPMD-GH-01-00097-01 at the Department of “Álgebra, Geometría y Topología” of the Valladolid University (Valladolid, Spain).
Communicated by: Michael Stillman
Article copyright: © Copyright 2004 American Mathematical Society

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