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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Linear systems of plane curves with base points of equal multiplicity
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by Ciro Ciliberto and Rick Miranda PDF
Trans. Amer. Math. Soc. 352 (2000), 4037-4050 Request permission

Abstract:

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
  • C. Ciliberto and R. Miranda: “Degenerations of Planar Linear Systems”, J. Reine Angew. Math. 501 (1998), 191–220.
  • A. Gimigliano: “On Linear Systems of Plane Curves”. Ph.D. Thesis, Queen’s University, Kingston, Ontario, Canada (1987).
  • Alessandro Gimigliano, Our thin knowledge of fat points, The Curves Seminar at Queen’s, Vol. VI (Kingston, ON, 1989) Queen’s Papers in Pure and Appl. Math., vol. 83, Queen’s Univ., Kingston, ON, 1989, pp. Exp. No. B, 50. MR 1036032
  • Brian Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane, Proceedings of the 1984 Vancouver conference in algebraic geometry, CMS Conf. Proc., vol. 6, Amer. Math. Soc., Providence, RI, 1986, pp. 95–111. MR 846019
  • André Hirschowitz, La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Math. 50 (1985), 337–388 (French, with English summary). MR 784148, DOI 10.1007/BF01168836
  • André Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles génériques, J. Reine Angew. Math. 397 (1989), 208–213 (French). MR 993223, DOI 10.1515/crll.1989.397.208
  • Morgan Ward, Note on the general rational solution of the equation $ax^2-by^2=z^3$, Amer. J. Math. 61 (1939), 788–790. MR 23, DOI 10.2307/2371337
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Additional Information
  • Ciro Ciliberto
  • Affiliation: Dipartimento of Mathematics, Universitá di Roma II, Via Fontanile di Carcaricola, 00173 Rome, Italy
  • MR Author ID: 49480
  • Email: cilibert@axp.mat.utovrm.it
  • Rick Miranda
  • Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
  • Email: miranda@math.colostate.edu
  • Received by editor(s): July 1, 1998
  • Published electronically: April 21, 2000
  • Additional Notes: Research supported in part by the NSA
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 4037-4050
  • MSC (1991): Primary 14H50, 14J26
  • DOI: https://doi.org/10.1090/S0002-9947-00-02416-8
  • MathSciNet review: 1637062