A classification of one-dimensional local domains based on the invariant (c-delta)r-delta

Let (R, m) be a one-dimensional, local, Noetherian domain and let R' be the integral closure of R in its quotient field K. We assume that R is not regular, analitycally irreducible and residually rational. The usual valuation v : K\longrightarrow Z \cup \infty associated to R' defines the numerical semigroup v (R)= { v (a), a in R, a\neq 0} \subseteq N. The aim of the paper is to study the non-negative invariant b:=(c- delta)r- delta , where c, delta, r denote respectively the conductor, the length of R'/R and the Cohen Macaulay type of R. In particular, the classification of the semigroups v(R) for rings R having b\leq 2(r-1) is realized. This method of classification might be successfully utilized with similar arguments but more boring computations in the cases b \leq q(r-1), for reasonably low values of q . The main tools are type sequences and the invariant k which estimates the number of elements in v(R) belonging to the interval [c-e,c-1], e being the multiplicity of R .