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Transactions of the American Mathematical Society

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



Linear systems of plane curves with base points of equal multiplicity

Authors: Ciro Ciliberto and Rick Miranda
Journal: Trans. Amer. Math. Soc. 352 (2000), 4037-4050
MSC (1991): Primary 14H50, 14J26
Published electronically: April 21, 2000
MathSciNet review: 1637062
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In this article we address the problem of computing the dimension of the space of plane curves of degree $d$with $n$ general points of multiplicity $m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple $(-1)$-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all $m \leq 12$.

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

  • [CM] C. Ciliberto and R. Miranda: ``Degenerations of Planar Linear Systems'', J. Reine Angew. Math. 501 (1998), 191-220. CMP 98:16
  • [G1] A. Gimigliano: ``On Linear Systems of Plane Curves''. Ph.D. Thesis, Queen's University, Kingston, Ontario, Canada (1987).
  • [G2] A. Gimigliano: ``Our thin knowledge of fat points'', in: Queen's papers in Pure and Applied Mathematics, vol. 83, The Curves Seminar at Queen's, Vol. VI, Queen's University, Kingston, Ontario, Canada (1989). MR 91a:14007
  • [Ha1] B. Harbourne: The Geometry of rational surfaces and Hilbert functions of points in the plane, Canad. Math. Soc. Conf. Proc., vol. 6 (1986), 95-111. MR 87k:14041
  • [Hi1] A. Hirschowitz: La méthode d'Horace pour l'interpolation à plusiers variables, Manuscripta Math., vol. 50 (1985), 337-388. MR 86j:14013
  • [Hi3] A. Hirschowitz: Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles generiques, J. Reine Angew. Math., vol. 397 (1989) 208-213. MR 90g:14021
  • [N] M. Nagata: On rational surfaces II, Memoirs of the College of Science, University of Kyoto, Series A, Vol. 33, Mathematics No. 2, (1960), 271-293. MR 23:93740

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Additional Information

Ciro Ciliberto
Affiliation: Dipartimento of Mathematics, Universitá di Roma II, Via Fontanile di Carcaricola, 00173 Rome, Italy

Rick Miranda
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523

Received by editor(s): July 1, 1998
Published electronically: April 21, 2000
Additional Notes: Research supported in part by the NSA
Article copyright: © Copyright 2000 American Mathematical Society

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