## Anticanonical Rational Surfaces

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- by Brian Harbourne PDF
- Trans. Amer. Math. Soc.
**349**(1997), 1191-1208 Request permission

## 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

- Michael Artin,
*Some numerical criteria for contractability of curves on algebraic surfaces*, Amer. J. Math.**84**(1962), 485–496. MR**146182**, DOI 10.2307/2372985 - F. Catanese,
*Pluricanonical Gorenstein curves*, preprint. - Ciro Ciliberto,
*On the degree and genus of smooth curves in a projective space*, Adv. Math.**81**(1990), no. 2, 198–248. MR**1055647**, DOI 10.1016/0001-8708(90)90009-C - Michel Demazure, Henry Charles Pinkham, and Bernard Teissier (eds.),
*Séminaire sur les Singularités des Surfaces*, Lecture Notes in Mathematics, vol. 777, Springer, Berlin, 1980 (French). Held at the Centre de Mathématiques de l’École Polytechnique, Palaiseau, 1976–1977. MR**579026** - Robert Friedman and David R. Morrison (eds.),
*The birational geometry of degenerations*, Progress in Mathematics, vol. 29, Birkhäuser, Boston, Mass., 1983. Based on papers presented at the Summer Algebraic Geometry Seminar held at Harvard University, Cambridge, Mass. June 11–July 29, 1981. MR**690261** - Brian Harbourne,
*Complete linear systems on rational surfaces*, Trans. Amer. Math. Soc.**289**(1985), no. 1, 213–226. MR**779061**, DOI 10.1090/S0002-9947-1985-0779061-2 - Brian Harbourne,
*Very ample divisors on rational surfaces*, Math. Ann.**272**(1985), no. 1, 139–153. MR**794097**, DOI 10.1007/BF01455934 - Brian Harbourne,
*Blowings-up of $\textbf {P}^2$ and their blowings-down*, Duke Math. J.**52**(1985), no. 1, 129–148. MR**791295**, DOI 10.1215/S0012-7094-85-05208-1 - Brian Harbourne,
*Automorphisms of $K3$-like rational surfaces*, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 17–28. MR**927971** - Brian Harbourne,
*Automorphisms of cuspidal $K3$-like surfaces*, Algebraic geometry: Sundance 1988, Contemp. Math., vol. 116, Amer. Math. Soc., Providence, RI, 1991, pp. 47–60. MR**1108631**, DOI 10.1090/conm/116/1108631 - Brian Harbourne,
*Rational surfaces with $K^2>0$*, Proc. Amer. Math. Soc.**124**(1996), no. 3, 727–733. MR**1307526**, DOI 10.1090/S0002-9939-96-03226-1 - Robin Hartshorne,
*Algebraic geometry*, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR**0463157**, DOI 10.1007/978-1-4757-3849-0 - Jean-Pierre Jouanolou,
*Théorèmes de Bertini et applications*, Progress in Mathematics, vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 (French). MR**725671** - Yu. I. Manin,
*Cubic forms*, 2nd ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. Algebra, geometry, arithmetic; Translated from the Russian by M. Hazewinkel. MR**833513** - Alan L. Mayer,
*Families of $K-3$ surfaces*, Nagoya Math. J.**48**(1972), 1–17. MR**330172**, DOI 10.1017/S002776300001504X - I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič,
*Torelli’s theorem for algebraic surfaces of type $\textrm {K}3$*, Izv. Akad. Nauk SSSR Ser. Mat.**35**(1971), 530–572 (Russian). MR**0284440** - B. Saint-Donat,
*Projective models of $K-3$ surfaces*, Amer. J. Math.**96**(1974), 602–639. MR**364263**, DOI 10.2307/2373709 - Fumio Sakai,
*Anticanonical models of rational surfaces*, Math. Ann.**269**(1984), no. 3, 389–410. MR**761313**, DOI 10.1007/BF01450701 - Hans Sterk,
*Finiteness results for algebraic $K3$ surfaces*, Math. Z.**189**(1985), no. 4, 507–513. MR**786280**, DOI 10.1007/BF01168156 - Tohsuke Urabe,
*On singularities on degenerate del Pezzo surfaces of degree $1,$ $2$*, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, R.I., 1983, pp. 587–591. MR**713283**

## Additional 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 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.
- © Copyright 1997 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**349**(1997), 1191-1208 - MSC (1991): Primary 14C20, 14J26; Secondary 14M20, 14N05
- DOI: https://doi.org/10.1090/S0002-9947-97-01722-4
- MathSciNet review: 1373636