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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Rational surfaces with $K^2>0$

Author: Brian Harbourne
Journal: Proc. Amer. Math. Soc. 124 (1996), 727-733
MSC (1991): Primary 14C20, 14J26; Secondary 13D40, 13P99
MathSciNet review: 1307526
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Abstract: The main but not all of the results in this paper concern rational surfaces $X$ for which the self-intersection $K_X^2$ of the anticanonical class $-K_X$ is positive. In particular, it is shown that no superabundant numerically effective divisor classes occur on any smooth rational projective surface $X$ with $K_X^2>0$. As an application, it follows that any 8 or fewer (possibly infinitely near) points in the projective plane $\mathbf{P}^2$ are in good position. This is not true for 9 points, and a characterization of the good position locus in this case is also given. Moreover, these results are put into the context of conjectures for generic blowings up of $\mathbf{P}^2$. All results are proven over an algebraically closed field of arbitrary characteristic.

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Additional Information

Brian Harbourne
Affiliation: Department of Mathematics and Statistics University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0323

Keywords: Anticanonical, rational, surface, good position
Received by editor(s): September 26, 1994
Additional Notes: This work was supported both by the National Science Foundation and by a Spring 1994 University of Nebraska Faculty Development Leave. I would like to thank Rick Miranda and Bruce Crauder for organizing the May 1994 Mtn. West Conference, where some of the results here were presented.
Communicated by: Eric Friedlander
Article copyright: © Copyright 1996 American Mathematical Society