A classification of one-dimensional local domains based on the invariant \((c-\delta)r-\delta\) (Q412532)

Let \((R, \mathfrak{m})\) be a one-dimensional, local, Noetherian domain and let \(\overline{R}\) be the integral closure of \(R\) in its quotient field \(K\). The authors assume that \(R\) is analytically irreducible (i.e., \(\overline{R}\) is a DVR and a finite \(R\)-module) and residually rational (i.e., \(R/\mathfrak{m}\cong \overline{R}/t\overline{R}\), where \(t \) denotes the uniformizing parameter of \(\overline{R}\)) but \(R\) is not a local regular ring. Let \( v : K \rightarrow \mathbb Z \cup \infty\) be the usual valuation associated to \(\overline{R}\), and let \(v(R) = \{v(a) \mid a \in R,\, a \neq 0\} \;(\subseteq \mathbb N)\) the associated numerical semigroup.NEWLINENEWLINEUsing the following classical invariants: \(c\), the conductor of \(R\) (i.e., the minimal natural integer \(j \in v( R)\) such that \( j + \mathbb N \subseteq v( R)\)), \(\delta := \ell_R (\overline{R}/R)\) (i.e., the number of ``gaps'' of the semigroup \(v( R)\) in \(\mathbb N\)), and \(r :=\ell_R((R : \mathfrak{m}))/R)\) (called the Cohen-Macaulay type of \(R\)), a new invariant \(b := (c - \delta)r -\delta\) has been recently considered in the literature and the general problem of classifying rings according to the ``size'' of \(b\) has been examined by several authors. In 1992, Brown and Herzog have characterized all the one-dimensional reduced local rings having \( b = 0 \) or \(b = 1\). Successively, D. Defino, M. D'Anna, V. Micale, L. Leer and R. Muntean in various papers have considered the rings for which \(b \leq r\). The present paper provides a classification of the semigroups \(v(R)\) for rings \(R\) having \(b\leq 2(r-1)\).