## 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

## 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

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