Cayley-Bacharach schemes and their canonical modules

Abstract

A set of s s points in P d {\mathbb {P}^d} is called a Cayley-Bacharach scheme ( CB {\text {CB}} -scheme), if every subset of s − 1 s - 1 points has the same Hilbert function. We investigate the consequences of this "weak uniformity." The main result characterizes CB {\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 CB {\text {CB}} -scheme X 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 CB {\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 CB {\text {CB}} -scheme, results about generic hyperplane sections of arithmetically Cohen-Macaulay curves and inequalities for the Hilbert functions of Cohen-Macaulay domains.