The Cohen-Macaulay type of points in generic position
Let \(X=\{P_ 1,...,P_ s\}\subset {\mathbb{P}}_ k^ n\), \(R=k[X_ 0,...,X_ n]\) and \(A=R/I\), where I is the ideal of X. The s points \(P_ i\) are said to be in generic position if the Hilbert function \(H_ A(t)\) satisfies \(H_ A(t)=\min \{s,H_ R(t)\}\) for all \(t\geq 0\). (This is not to be confused with ``in general position'', which means that no subset of \(n+1\) points lies on a hyperplane.) The ring A is a one-dimensional graded Cohen-Macaulay ring. In 1981, L. G. Roberts gave an explicit lower bound for the Cohen-Macaulay type \(\tau\) (A) of A for points in generic position, and conjectured that for a general set of points in generic position this lower bound would be actually attained. In the present paper the authors prove this conjecture by showing that, if \(P_ 1,...,P_ s\) are independent generic points (i.e. there coordinates are independent variables over k), then \(\tau\) (A) attains the expected value. The proof consists of a careful evaluation of the rank of a certain matrix the entries of which are monomials.
scientific article; zbMATH DE number 4148291