Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • 1. E. BALLICO, J. MIGLIORE, Smooth curves whose hyperplane section is a given set of points, Comm. Algebra 18 (1990), no. 9, 3015-3040. MR 1063348 (91h:14034)
  • 2. N. BUDUR, M. CASANELLAS, E. GORLA, Hilbert functions of integral standard determinantal schemes and integral arithmetically Gorenstein schemes, Journal of Algebra 272 (2004), 292-310. MR 2029035 (2004j:14054)
  • 3. A. CAPANI, G. NIESI, L. ROBBIANO, CoCoA, a system for doing Computations in Commutative Algebra, Available via anonymous ftp from:
  • 4. D. EISENBUD, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, Graduate Texts in Mathematics 150 (1995). MR 1322960 (97a:13001)
  • 5. G. GAETA, Nuove ricerche sulle curve sghembe algebriche di residuale finito e sui gruppi di punti del piano, Ann. Mat. Pura ed Appl. (4) 31 (1950), 1-64. MR 0042744 (13:156c)
  • 6. A.V. GERAMITA, J. MIGLIORE, On the ideal of an arithmetically Buchsbaum curve, J.P.A.A. 54 (1988), 215-247. MR 0963546 (90d:14053)
  • 7. A.V. GERAMITA, J. MIGLIORE, Generators for the ideal of an Arithmetically Buchsbaum curve, J.P.A.A. 58 (1989), 147-167.MR 1001472 (90f:14017)
  • 8. A.V. GERAMITA, J. MIGLIORE, Hyperplane Sections of a Smooth Curve in $\mathbf{P}^3$, Comm. Algebra 17 (1989), 3129-3164. MR 1030613 (90k:14027)
  • 9. S. GRECO, P. VALABREGA, On the Singular Locus of a general complete intersection through a Variety in Projective Space, B.U.M.I. Alg. e Geom. VI, v.II, n. 1 (1983), 113-145. MR 0771535 (86j:14049)
  • 10. L. GRUSON, C. PESKINE, Section plane d'une courbe gauche: postulation Enumerative geometry and classical algebraic geometry, (Nice, 1981), Progr. Math., 24, Birkhäuser, Boston, Mass, (1982), 33-35.MR 0685762 (84c:14045)
  • 11. J. HARRIS, The genus of space curves, Math. Ann. 249 (1980), 191-204.MR 0579101 (81i:14022)
  • 12. M. HERMANN, S. IKEDA AND U. ORBANZ, Equimultiplicity and Blowing Up, Springer-Verlag, Berlin (1988). MR 0954831 (89g:13012)
  • 13. J. HERZOG, N.V. TRUNG AND G. VALLA, Hyperplane sections of reduced irreducible varieties of low codimension, J. Math. Kyoto Univ. 33 (1994), 47-72. MR 1263860 (95d:14048)
  • 14. C. HUNEKE, B. ULRICH, General hyperplane sections of Algebraic Varieties, J. Alg. Geom. 2 (1993), 487-505. MR 1211996 (94b:14046)
  • 15. J.O. KLEPPE, J.C. MIGLIORE, R. MIRÓ-ROIG, U. NAGEL, C. PETERSON, Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness, Memoirs AMS 732 (2001), vol. 154. MR 1848976 (2002e:14083)
  • 16. M. KREUZER, J. MIGLIORE, U. NAGEL, C. PETERSON, Determinantal schemes and Buchsbaum-Rim sheaves, J.P.A.A. 150 (2000), no. 2, 155-174.MR 1765869 (2001f:14092)
  • 17. O.A. LAUDAL, A generalized trisecant Lemma, Algebraic Geometry, Lecture Notes in Math., vol. 687, Springer-Verlag, Berlin, Heidelberg and New York (1987), 112-149. MR 0527232 (81f:14019)
  • 18. R. MAGGIONI, A. RAGUSA, The Hilbert function of generic plane sections of curves of $\mathbf{P}^3$, Invent. Math. 91 (1988), no. 2, 253-258.MR 0922800 (89g:14027)
  • 19. M. MARTIN-DESCHAMPS, D. PERRIN, Construction de courbes lisses: un Theorem à la Bertini, Laboratoire de Mathématiques École Normale Supérieure (LMENS) 22 (1992).
  • 20. J. MIGLIORE, Introduction to Liaison Theory and Deficiency Modules, Birkhäuser, Progress in Mathematics 165 (1998). MR 1712469 (2000g:14058)
  • 21. J. MIGLIORE, Submodules of the Deficiency Module, J. London Math. Soc. (2) 48 (1993), 396-414. MR 1241777 (94i:14051)
  • 22. J. MIGLIORE, Hypersurface Sections of Curves, Zero-dimensional schemes (Ravello, 1992), 269-282, de Gruyter, Berlin (1994). MR 1292491 (95i:14043)
  • 23. J. MIGLIORE, U. NAGEL, C. PETERSON, Buchsbaum-Rim sheaves and their multiple sections, J. Algebra 219 (1999), no. 1, 378-420.MR 1707678 (2000f:14076)
  • 24. S.R. NOLLET, Integral curves in even linkage classes, Ph.D. Thesis, University of California at Berkeley.
  • 25. G. PAXIA, A. RAGUSA, Irreducible Buchsbaum Curves, Comm. Alg. 23, n. 8 (1995), 3025-3031. MR 1332162 (96f:14037)
  • 26. R. RE, Sulle sezioni iperpiane di una Varietá Proiettiva, Le Matematiche 42 (1987), 211-218.
  • 27. T. SAUER, Smoothing Projectively Cohen-Macaulay Space Curves, Math. Ann. 272 (1985), 83-90. MR 0794092 (87c:14060)
  • 28. P. SCHWARTAU, Liaison Addition and Monomial Ideals, Ph.D. Thesis, Brandeis University (1982).
  • 29. R. STRANO, A characterization of Complete Intersection Curves in $\mathbf{P}^3$, Proc. Amer. Math. Soc. 104 (1988), 711-715.MR 0964847 (90b:14062)
  • 30. J. STÜCKRAD, W. VOGEL, Buchsbaum Rings and Applications, VEB Deutscher Verlag der Wissenschaften (1986). MR 0881220 (88h:13011a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14H99, 14M05, 13F20

Retrieve articles in all journals with MSC (2000): 14H99, 14M05, 13F20

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

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

American Mathematical Society