Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Anticanonical Rational Surfaces

Author: Brian Harbourne
Journal: Trans. Amer. Math. Soc. 349 (1997), 1191-1208
MSC (1991): Primary 14C20, 14J26; Secondary 14M20, 14N05
MathSciNet review: 1373636
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically closed field of arbitrary characteristic. Applications, to be treated in separate papers, include questions involving: points in good position, birational models of rational surfaces in projective space, and resolutions for 0-dimensional subschemes of $\mathbf {P}^{2}$ defined by complete ideals.

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
  • [Cn] F. Catanese, Pluricanonical Gorenstein curves, preprint.
  • [Co] C. Ciliberto, On the degree and genus of smooth curves in a projective space, Adv. Math. 81 (1990), 198-248. MR 91h:14035
  • [D] M. Demazure, Surfaces de Del Pezzo - IV. Systèmes anticanoniques, Séminaire sur les Singularités des Surfaces, LNM, vol. 777, 1980. MR 82d:14021
  • [F] R. Friedman, Appendix: Linear systems on anticanonical pairs, The Birational Geometry of Degenerations, vol. 29, Prog. Math., Birkhäuser, Boston, 1983, pp. 162-171. MR 84a:14001
  • [H1] B. Harbourne, Complete linear systems on rational surfaces, Trans. A. M. S. 289 (1985), 213-226. MR 86h:14030
  • [H2] -, Very ample divisors on rational surfaces, Math. Ann. 272 (1985), 139-153. MR 86k:14026
  • [H3] -, Blowings-up of $\mathbf {P}^{2}$ and their blowings-down, Duke Math. J. 52 (1985), 129-148. MR 86m:14026
  • [H4] -, Automorphisms of $K3$-like rational surfaces, Proc. Symp. Pure Math., vol. 46, 1987, pp. 17-28. MR 89f:14044
  • [H5] -, Automorphisms of cuspidal K3-like surfaces, Algebraic Geometry: Sundance 1988, Contemporary Mathematics, vol. 116, 1991, pp. 47-60. MR 92c:14039
  • [H6] -, Rational surfaces with $K^{2}>0$, Proc. A.M.S. 124 (1996), 727-733. MR 96f:14045
  • [Ha] R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977. MR 57:3116
  • [J] J-P. Jouanolou, Theoremes de Bertini et applications., Prog. Math., vol. 42, Birkhäuser, Boston, 1983. MR 86b:13007
  • [M] Y. I. Manin, Cubic Forms, North-Holland Mathematical Library, vol. 4, 1986. MR 87d:11037
  • [Ma] A. Mayer, Families of K3 surfaces, Nagoya J. Math. 48 (1972), 1-17. MR 48:8510
  • [PS] I. Piateckii-Shapiro and I. R. Shafarevich, A Torelli theorem for algebraic surfaces of type $K3$, Math. USSR Izv. 5 (1971), 547-588. MR 44:1666
  • [SD] B. Saint-Donat, Projective models of $K3$ surfaces, Amer. J. Math. 96 (1974), 602-639. MR 51:518
  • [Sk] F. Sakai, Anticanonical models of rational surfaces, Math. Ann. 269 (1984), 389-410. MR 85m:14058
  • [St] H. Sterk, Finiteness results for algebraic $K3$ surfaces, Math. Z. 189 (1985), 507-513. MR 86j:14038
  • [U] T. Urabe, On singularities on degenerate Del Pezzo surfaces of degree $1$, $2$, Proc. Symp. Pure Math., vol. 40(2), 1983, pp. 587-592. MR 84i:14024

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 14C20, 14J26, 14M20, 14N05

Retrieve articles in all journals with MSC (1991): 14C20, 14J26, 14M20, 14N05

Additional Information

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

Keywords: Anticanonical, rational, surface, base points, fixed components, linear systems
Received by editor(s): September 29, 1995
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 also like to thank Tony Geramita for a helpful discussion, and the referee for a careful reading of the paper.
Article copyright: © Copyright 1997 American Mathematical Society