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)



Cayley-Bacharach schemes and their canonical modules

Authors: Anthony V. Geramita, Martin Kreuzer and Lorenzo Robbiano
Journal: Trans. Amer. Math. Soc. 339 (1993), 163-189
MSC: Primary 14M05; Secondary 13D40
MathSciNet review: 1102886
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A set of $ s$ points in $ {\mathbb{P}^d}$ is called a Cayley-Bacharach scheme ( $ {\text{CB}}$-scheme), if every subset of $ s - 1$ points has the same Hilbert function. We investigate the consequences of this "weak uniformity." The main result characterizes $ {\text{CB}}$-schemes in terms of the structure of the canonical module of their projective coordinate ring. From this we get that the Hilbert function of a $ {\text{CB}}$-scheme $ X$ has to satisfy growth conditions which are only slightly weaker than the ones given by Harris and Eisenbud for points with the uniform position property. We also characterize $ {\text{CB}}$-schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points. Applications include efficient algorithms for checking whether a given set of points is a $ {\text{CB}}$-scheme, results about generic hyperplane sections of arithmetically Cohen-Macaulay curves and inequalities for the Hilbert functions of Cohen-Macaulay domains.

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

  • [BV] W. Bruns and U. Vetter, Determinantal rings, Lecture Notes in Math., vol. 1327, Springer-Verlag, Heidelberg, 1988. MR 953963 (89i:13001)
  • [C] G. Castelnuovo, Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica, Rend. Circ. Mat. Palermo 7 (1893), 89-110.
  • [CO] L. Chiantini and F. Orecchia, Plane sections of arithmetically normal curves in $ {\mathbb{P}^3}$, Algebraic Curves and Projective Geometry, Proceedings, Trento 1988, Lecture Notes in Math., vol. 1389, Springer-Verlag, Heidelberg, 1989, pp. 32-42. MR 1023388 (90j:14038)
  • [D] E. D. Davis, Open problems, The Curves Seminar at Queen's, vol. III, Queen's Papers in Pure and Appl. Math. no. 67, Queen's University, Kingston, 1984, p. 8. MR 783095 (86g:14027)
  • [DGM] E. D. Davis, A. V. Geramita, and P. Maroscia, Perfect homogeneous ideals: Dubreil's theorem revisited, Bull. Sci. Math. (2) 108 (1984), 143-185. MR 769926 (86m:13024)
  • [DGO] E. D. Davis, A. V. Geramita, and F. Orecchia, Gorenstein algebras and the Cayley-Bacharach theorem, Proc. Amer. Math. Soc. 93 (1985), 593-597. MR 776185 (86k:14034)
  • [G] W. Gröbner, Über irreduzible Ideale in kommutativen Ringen, Math. Ann. 110 (1934), 197-222.
  • [GM] A. V. Geramita and J. C. Migliore, Hyperplane sections of a smooth curve in $ {\mathbb{P}^3}$, Comm. Algebra 17 (1989), 3129-3164. MR 1030613 (90k:14027)
  • [GMR] A. V. Geramita, P. Maroscia, and L. G. Roberts, The Hilbert function of a reduced $ k$-algebra, J. London Math. Soc. (2) 28 (1983), 443-452. MR 724713 (85c:13018)
  • [GN] A. Giovini and G. Niesi, CoCoA user's manual, v. 099b, Dipartimento di Matematica, Università di Genova, Genova, 1989.
  • [GW] S. Goto and K. Watanabe, On graded rings. I, J. Math. Soc. Japan 30 (1978), 179-213. MR 494707 (81m:13021)
  • [H] J. Harris, The genus of space curves, Math. Ann. 249 (1980), 191-204. MR 579101 (81i:14022)
  • [HE] J. Harris (with the collaboration of D. Eisenbud), Curves in projective space, Sém. de Mathématiques Supérieures, Université de Montreal, 1982. MR 685427 (84g:14024)
  • [K1] M. Kreuzer, On 0-dimensional complete intersections, preprint, The Curves Seminar at Queen's, vol. VII, Queen's Papers in Pure and Appl. Math., no. 85, Queen's University, Kingston, 1990. MR 1089903 (92c:14050)
  • [K2] -, Vektorbündel und der Satz von Cayley-Bacharach, Dissertation, Regensburger Mathematische Schriften 21, Universität Regensburg, 1989. MR 1027141 (91d:14035)
  • [KW] E. Kunz and R. Waldi, Regular differential forms, Contemporary Math., vol. 79, Amer. Math. Soc., Providence, R.I., 1988. MR 971502 (90a:14021)
  • [O] F. Orecchia, Points in generic position and the conductor of curves with ordinary singularities, J. London Math. Soc. (2) 24 (1981), 85-96. MR 623673 (82j:14024)
  • [Ra] J. Rathmann, The uniform position principle for curves in characteristic $ p$, Math. Ann. 276 (1987), 565-579. MR 879536 (89g:14026)
  • [Ro] L. Robbiano, Introduction to the theory of Gröbner bases, The Curves Seminar at Queen's, vol. V, Queen's Papers in Pure and Appl. Math., no. 80, Queen's University, Kingston, 1988, B1-B29. MR 973648 (90a:13003)
  • [So] A. Sodhi, On the conductor of points in $ {\mathbb{P}^n}$, Dissertation, Queen's University, Kingston, 1987.
  • [St] R. Stanley, On the Hilbert function of a graded Cohen-Macaulay domain, preprint, Massachusetts Inst. of Technology, Cambridge, 1990. MR 1124790 (92f:13017)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 14M05, 13D40

Retrieve articles in all journals with MSC: 14M05, 13D40

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society