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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

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Rational surfaces with $K^2>0$
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by Brian Harbourne
Proc. Amer. Math. Soc. 124 (1996), 727-733
DOI: https://doi.org/10.1090/S0002-9939-96-03226-1
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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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Bibliographic Information
  • Brian Harbourne
  • Affiliation: Department of Mathematics and Statistics University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0323
  • MR Author ID: 217048
  • Email: bharbourne@unl.edu
  • 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
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 727-733
  • MSC (1991): Primary 14C20, 14J26; Secondary 13D40, 13P99
  • DOI: https://doi.org/10.1090/S0002-9939-96-03226-1
  • MathSciNet review: 1307526