Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-96-03226-1
MathSciNet review: 1307526
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


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

  • [A] M. Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485--497. MR 26:3704
  • [D] M. Demazure, Surfaces de Del Pezzo - II. Eclater $n$ points de $\hbox {\strut \bf P}^{\strut 2}$, Séminaire sur les Singularités des Surfaces, Lecture Notes in Math., vol. 777, Springer-Verlag, New York, 1980.
  • [Gi] A. Gimigliano, Our thin knowledge of fat points, The Curves Seminar at Queen's, vol. VI, Queen's Papers in Pure and Appl. Math., vol. 83, Queen's Univ., Kingston, Ontario (1989). MR 91a:14007
  • [H1] B. Harbourne, Complete linear systems on rational surfaces, Trans. Amer. Math. Soc. 289 (1985), 213--226. MR 86h:14030
  • [H2] B. Harbourne, Blowings-up of $\hbox {\strut \bf P}^{\strut 2}$ and their blowings-down, Duke Math. J. 52 (1985), 129--148. MR 86m:14026
  • [H3] B. Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane, Canad. Math. Soc. Conf. Proc. 6 (1986), 95--111. MR 87k:14041
  • [H4] B. Harbourne, Iterated blow-ups and moduli for rational surfaces, Algebraic Geometry: Sundance 1986, Lecture Notes in Math., vol. 1311 , Springer-Verlag, New York, 1988, pp. 101--117. MR 90b:14009
  • [H5] B. Harbourne, Points in good position in $\hbox {\strut \bf P}^{\strut 2}$, Zero-dimensional schemes, Proceedings of the International Conference held in Ravello, Italy, June 8--13, 1992, De Gruyter, Berlin, 1994, CMP 94:17.
  • [H6] B. Harbourne, Anticanonical rational surfaces, preprint 1994.
  • [Ha] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. MR 57:3116
  • [Hi] A. Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles generiques, J. Reine Angew. Math. 397 (1989), 208--213. MR 90g:14021
  • [R] C. P. Ramanujam, Supplement to the article ``Remarks on the Kodaira vanishing theorem'', J. Indiana Math. Soc. 38 (1974), 121--124. MR 52:13859

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 14C20, 14J26, 13D40, 13P99

Retrieve articles in all journals with MSC (1991): 14C20, 14J26, 13D40, 13P99


Additional Information

Brian Harbourne
Affiliation: Department of Mathematics and Statistics University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0323
Email: bharbourne@unl.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03226-1
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

American Mathematical Society