Cohen-Macaulayness of almost complete intersection tangent cones
The author considers affine monomial curves of codimension three, that is, curves which coordinate ring is \(K[[T^{n_1},..., T^{n_1}]]\), where \(n_1,...,n_4\) are nonzero natural integers. A monomial curve can be seen as a quotient \(K[X,Y,Z,W]/I\simeq K[[T^{n_1},..., T^{n_1}]]\), where \(I\) is generated by some binomials. The tangent cone is defined by the ideal \(I^*\subset K[X,Y,Z,W]\) generated by the initial forms of all elements in \(I,\) for the usual degree on \(K[X,Y,Z,W]\). The main theorem in this paper is the following: If \(K[[T^{n_1},..., T^{n_1}]]\) is a complete intersection and if the ideal \(I^*\) defining the tangent cone at the origin is generated by 4 elements then the tangent cone is arithmetically Cohen-Macaulay. The proof consist to give an algorithm to compute a minimal basis of \(I^*\), by extending some results on the theory of standard basis of modules, similar to the theory Gröbner basis.
scientific article; zbMATH DE number 5300805
scientific article