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Irreducibility of the $(-1)$-classes on smooth rational surfaces


Author: Mustapha Lahyane
Journal: Proc. Amer. Math. Soc. 133 (2005), 2219-2224
MSC (2000): Primary 14J26; Secondary 14F05
DOI: https://doi.org/10.1090/S0002-9939-05-07654-9
Published electronically: March 14, 2005
MathSciNet review: 2138862
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Abstract: We give a characterization for a $(-1)$-divisor $D$ on a smooth rational surface $X$ to be irreducible under the assumption that an anticanonical divisor $-K_X$ of $X$ is nef. Here $-K_X$ is nef means $K_X . C \leq 0$ for every effective divisor $C$ on $X$, and a $(-1)$-divisor $D$ is a divisor such that the two numerical conditions $D^2 =-1=D.K_X$ hold.

As an application we give explicit examples of blowing up the projective plane at nine points infinitely near such that the obtained surface has an infinite number of $(-1)$-curves. A $(-1)$-curve is a smooth rational curve of self-intersection $-1$.


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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 Àlgebra, Geometrìa y Topologìa, Facultad de Ciencias, 47005 Valladolid, Spain
Email: lahyane@agt.uva.es

DOI: https://doi.org/10.1090/S0002-9939-05-07654-9
Keywords: Smooth rational surfaces, anticanonical divisor, Hodge Index Theorem, points in general position, N\'eron-Severi group, blowing-up.
Received by editor(s): June 28, 2001
Received by editor(s) in revised form: October 15, 2003
Published electronically: March 14, 2005
Additional Notes: The author was partially supported by a postdoctoral fellowship at the International Centre for Theoretical Physics (Trieste, Italy) and a Marie Curie grant number HPMD-GH-01-00097-01 at the department of “Álgebra, Geometría y Topología” of Valladolid University (Valladolid, Spain).
Dedicated: This research is dedicated to my mother, Hnia Hamami
Communicated by: Michael Stillman
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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