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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 a composite number of base points of equal multiplicity


Authors: Anita Buckley and Marina Zompatori
Journal: Trans. Amer. Math. Soc. 355 (2003), 539-549
MSC (2000): Primary 14H50
Published electronically: October 1, 2002
MathSciNet review: 1932712
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Abstract: In this article we study linear systems of plane curves of degree $d$ passing through general base points with the same multiplicity at each of them. These systems are known as homogeneous linear systems. We especially investigate for which of these systems, the base points, with their multiplicities, impose independent conditions and which homogeneous systems are empty. Such systems are called non-special. We extend the range of homogeneous linear systems that are known to be non-special. A theorem of Evain states that the systems of curves of degree $d$ with $4^h$ base points with equal multiplicity are non-special. The analogous result for $9^h$ points was conjectured. Both of these will follow, as corollaries, from the main theorem proved in this paper. Also, the case of $4^{h}9^{k}$ points will follow from our result. The proof uses a degeneration technique developed by C. Ciliberto and R. Miranda.


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

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Additional Information

Anita Buckley
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: mocnik@maths.warwick.ac.uk

Marina Zompatori
Affiliation: Department of Mathematics, Boston University, Boston, Massachusetts 02215
Email: marinaz@math.bu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03164-1
PII: S 0002-9947(02)03164-1
Received by editor(s): February 25, 2002
Published electronically: October 1, 2002
Additional Notes: We wish to thank the organizers of Pragmatic 2001 for sponsoring our stay in Catania, Rick Miranda and Ciro Ciliberto for introducing us to problems on linear systems and for many invaluable conversations. We would also like to thank Dan Abramovich and Balázs Szendrői for corrections and helpful comments.
Article copyright: © Copyright 2002 American Mathematical Society