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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
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.

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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

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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