Degeneration of linear systems through fat points on $K3$ surfaces
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- by Cindy De Volder and Antonio Laface PDF
- Trans. Amer. Math. Soc. 357 (2005), 3673-3682 Request permission
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 $\mathbb {P}^3$.References
- J. Alexander and A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math. 140 (2000), no. 2, 303–325. MR 1756998, DOI 10.1007/s002220000053
- Anita Buckley and Marina Zompatori, Linear systems of plane curves with a composite number of base points of equal multiplicity, Trans. Amer. Math. Soc. 355 (2003), no. 2, 539–549. MR 1932712, DOI 10.1090/S0002-9947-02-03164-1
- Ciro Ciliberto and Rick Miranda, Degenerations of planar linear systems, J. Reine Angew. Math. 501 (1998), 191–220. MR 1637857
- Ciro Ciliberto, Friedrich Hirzebruch, Rick Miranda, and Mina Teicher (eds.), Applications of algebraic geometry to coding theory, physics and computation, NATO Science Series II: Mathematics, Physics and Chemistry, vol. 36, Kluwer Academic Publishers, Dordrecht, 2001. MR 1866890, DOI 10.1007/978-94-010-1011-5
- Cindy De Volder and Antonio Laface. Linear systems on generic $K3$ surfaces. Preprint, math.AG/0309073, 2003.
- Evain Laurent, La fonction de Hilbert de la réunion de $4^h$ gros points génériques de $\textbf {P}^2$ de même multiplicité, J. Algebraic Geom. 8 (1999), no. 4, 787–796 (French, with French summary). MR 1703614
- Alessandro Gimigliano, Regularity of linear systems of plane curves, J. Algebra 124 (1989), no. 2, 447–460. MR 1011606, DOI 10.1016/0021-8693(89)90142-7
- Brian Harbourne, Complete linear systems on rational surfaces, Trans. Amer. Math. Soc. 289 (1985), no. 1, 213–226. MR 779061, DOI 10.1090/S0002-9947-1985-0779061-2
- Alan L. Mayer, Families of $K-3$ surfaces, Nagoya Math. J. 48 (1972), 1–17. MR 330172
- Joaquim Roé. On the Nagata conjecture. Preprint, math.AG/0304124, 2003.
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
- MR Author ID: 634848
- Email: antonio.laface@unimi.it
- 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) - © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2146644