Cayley-Bacharach schemes and their canonical modules

Geramita, Anthony V.; Kreuzer, Martin; Robbiano, Lorenzo

doi:10.1090/S0002-9947-1993-1102886-5

Cayley-Bacharach schemes and their canonical modules
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by Anthony V. Geramita, Martin Kreuzer and Lorenzo Robbiano

Trans. Amer. Math. Soc. 339 (1993), 163-189

DOI: https://doi.org/10.1090/S0002-9947-1993-1102886-5

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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.

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