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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Conditions imposed by tacnodes and cusps


Author: Joaquim Roé
Journal: Trans. Amer. Math. Soc. 353 (2001), 4925-4948
MSC (1991): Primary 14C20; Secondary 14H20, 14J26, 14H50, 14C05
Published electronically: April 18, 2001
MathSciNet review: 1852087
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Abstract:

The study of linear systems of algebraic plane curves with fixed imposed singularities is a classical subject which has recently experienced important progress. The Horace method introduced by A. Hirschowitz has been successfully exploited to prove many $H^1$-vanishing theorems, even in higher dimension. Other specialization techniques, which include degenerations of the plane, are due to Z. Ran and C. Ciliberto and R. Miranda. G. M. Greuel, C. Lossen and E. Shustin use a local specialization procedure together with the Horace method to give the first asymptotically proper general existence criterion for singular curves of low degree. In this paper we develop a specialization method which allows us to compute the dimension of several linear systems as well as to substantially improve the bounds given by Greuel, Lossen and Shustin for curves with tacnodes and cusps.


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Additional Information

Joaquim Roé
Affiliation: Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via, 585, E-08007, Barcelona
Email: jroevell@cerber.mat.ub.es

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02740-4
PII: S 0002-9947(01)02740-4
Received by editor(s): July 5, 1999
Received by editor(s) in revised form: April 13, 2000
Published electronically: April 18, 2001
Additional Notes: Partially supported by CIRIT 1997FI-00141, CAICYT PB95-0274, and “AGE-Algebraic Geometry in Europe" contract no.\ ERB940557
Article copyright: © Copyright 2001 American Mathematical Society