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Transactions of the American Mathematical Society
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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Abstract | References | Similar Articles | Additional Information

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

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Additional Information

Ciro Ciliberto
Affiliation: Dipartimento of Mathematics, Universitá di Roma II, Via Fontanile di Carcaricola, 00173 Rome, Italy
Email: cilibert@axp.mat.utovrm.it

Rick Miranda
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
Email: miranda@math.colostate.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02416-8
PII: S 0002-9947(00)02416-8
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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