The general hyperplane section of a curve
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Abstract:
In this paper, we discuss some necessary and sufficient conditions for a curve to be arithmetically Cohen-Macaulay, in terms of its general hyperplane section. We obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non-arithmetically Cohen-Macaulay curve of $\mathbf {P}^3$. We prove that almost all the degree matrices with positive subdiagonal that occur for the general plane section of a non-arithmetically Cohen-Macaulay curve of $\mathbf {P}^3$, arise also as degree matrices of some smooth, integral, non-arithmetically Cohen-Macaulay curve, and we characterize the exceptions. We give a necessary condition on the graded Betti numbers of the general plane section of an arithmetically Buchsbaum (non-arithmetically Cohen-Macaulay) curve in $\mathbf {P}^n$. For curves in $\mathbf {P}^3$, we show that any set of Betti numbers that satisfies that condition can be realized as the Betti numbers of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay curve. We also show that the matrices that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, integral, (smooth) non-arithmetically Cohen-Macaulay space curve are exactly those that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay space curve and have positive subdiagonal. We also prove some bounds on the dimension of the deficiency module of an arithmetically Buchsbaum space curve in terms of the degree matrix of the general plane section of the curve, and we prove that they are sharp.References
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Additional Information
- Elisa Gorla
- Affiliation: Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, Indiana 46556-4618
- Address at time of publication: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
- Email: egorla@nd.edu, elisa.gorla@math.unizh.ch
- Received by editor(s): June 3, 2003
- Received by editor(s) in revised form: April 12, 2004
- Published electronically: April 13, 2005
- Additional Notes: The author was partially supported by a scholarship from the Italian research institute “Istituto Nazionale di Alta Matematica Francesco Severi”
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 819-869
- MSC (2000): Primary 14H99, 14M05, 13F20
- DOI: https://doi.org/10.1090/S0002-9947-05-03701-3
- MathSciNet review: 2177042