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

 

The general hyperplane section of a curve


Author: Elisa Gorla
Journal: Trans. Amer. Math. Soc. 358 (2006), 819-869
MSC (2000): Primary 14H99, 14M05, 13F20
Published electronically: April 13, 2005
MathSciNet review: 2177042
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03701-3
PII: S 0002-9947(05)03701-3
Keywords: Arithmetically Cohen-Macaulay curve, general hyperplane section, degree matrix, lifting matrix, smooth and integral curve, arithmetically Buchsbaum curve, deficiency module
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”
Article copyright: © Copyright 2005 American Mathematical Society