A matrix computation for the Cohen-Macaulay type of s-lines in affine \((n+1)\)-space
A system of s distinct lines \(L_ 1,...L_ s\), passing through the origin in the affine space \({\mathbb{A}}_ k^{n+1}\) (k denoting an algebraically closed field), defines a simple curve singularity. The problem of computing the CM type \(t(L_ 1,...L_ s)\) of the local ring at the origin was considered by \textit{A. V. Geramita} and \textit{F. Orecchia} [J. Algebra 70, 116-140 (1981; Zbl 0464.14007)], who mostly dwelled on the case when the ''direction numbers'' \(P_ 1,...P_ s\) of those lines, considered as points of \({\mathbb{P}}^ n\), are in generic s- position. By definition, this means that the \(s\times (_ n^{d+n})\) matrix G(d), obtained by evaluating all degree d monomials on \(n+1\) indeterminates at the homogeneous coordinates of \(P_ 1,...,P_ s\), has maximal possible rank for every \(d\geq 1.-\) The present authors obtain a general formula in the form: \(t(L_ 1,...L_ s)=\sum^{s- 1}_{j=r}\gamma (j)+rk G(s-1)-(_{\quad n}^{n+s-2})\) where the \(\gamma\) (j)'s are - in principle - computable from certain matrices obtained from G by row-transformations (one has also to assume the leading entries of the direction numbers equal to 1, but this is no restriction on the generality); the integer r is the minimal degree of s hypersurface passing through all s points; \(s\geq 2\) and \(n\geq 2.\) This is used to deduce a useful expression in the case of generic s- position: \(t(L_ 1,...,L_ s)=\gamma (r)+s-(_{\quad n}^{r+n-1}).\) Examples are given to show that for s and n fixed, the type can vary even if the lines are in generic s-position.
scientific article; zbMATH DE number 3873489