On the Hilbert function of one-dimensional local complete intersections

The sequences that occur as Hilbert functions of standard graded algebras A are well understood by Macaulayʼs theorem; those that occur for graded complete intersections are elementary and were known classically. However, much less is known in the local case, once the dimension of A is greater than zero, or the embedding dimension is three or more.Using an extension to the power series ring R of Gröbner bases with respect to local degree orderings, we characterize the Hilbert functions H of one-dimensional quadratic complete intersections A=R/I, I=(f,g), of type (2,2) that is, that are quotients of the power series ring R in three variables by a regular sequence f, g whose initial forms are linearly independent and of degree 2. We also give a structure theorem up to analytic isomorphism of A for the minimal system of generators of I, given the Hilbert function.More generally, when the type of I is (2,b) we are able to give some restrictions on the Hilbert function. In this case we can also prove that the associated graded algebra of A is Cohen-Macaulay if and only if the Hilbert function of A is strictly increasing.