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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Cayley-Bacharach schemes and their canonical modules
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by Anthony V. Geramita, Martin Kreuzer and Lorenzo Robbiano PDF
Trans. Amer. Math. Soc. 339 (1993), 163-189 Request permission


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.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 339 (1993), 163-189
  • MSC: Primary 14M05; Secondary 13D40
  • DOI:
  • MathSciNet review: 1102886