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Degeneration of linear systems through fat points on $K3$ surfaces


Authors: Cindy De Volder and Antonio Laface
Journal: Trans. Amer. Math. Soc. 357 (2005), 3673-3682
MSC (2000): Primary 14C20, 14J28.
DOI: https://doi.org/10.1090/S0002-9947-04-03653-0
Published electronically: December 28, 2004
MathSciNet review: 2146644
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Abstract: In this paper we introduce a technique to degenerate $K3$surfaces and linear systems through fat points in general position on $K3$surfaces. Using this degeneration we show that on generic $K3$ surfaces it is enough to prove that linear systems with one fat point are non-special in order to obtain the non-speciality of homogeneous linear systems through $n = 4^u9^w$fat points in general position. Moreover, we use this degeneration to obtain a result for homogeneous linear systems through $n = 4^u9^w$ fat points in general position on a general quartic surface in $\mathbb{P}^3$.


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Additional Information

Cindy De Volder
Affiliation: Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Ghent, Belgium
Email: cdv@cage.ugent.be

Antonio Laface
Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20100 Milano, Italy
Email: antonio.laface@unimi.it

DOI: https://doi.org/10.1090/S0002-9947-04-03653-0
Keywords: Linear systems, fat points, generic $K3$ surfaces
Received by editor(s): October 24, 2003
Received by editor(s) in revised form: January 14, 2004
Published electronically: December 28, 2004
Additional Notes: The first author is a Postdoctoral Fellow of the Fund for Scientific Research-Flanders (Belgium) (F.W.O.-Vlaanderen)
The second author would like to thank the European Research and Training Network EAGER for the support provided at Ghent University. He also acknowledges the support of the MIUR of the Italian Government in the framework of the National Research Project “Geometry in Algebraic Varieties” (Cofin 2002)
Article copyright: © Copyright 2004 American Mathematical Society

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