Partition of complement of good ideals and Apéry sets

Abstract

Good semigroups form a class of submonoids of ${\mathbb{N}}^d$ containing the value semigroups of curve singularities. In this article, we describe a partition of the complements of good semigroup ideals, having as main application the description of the Apéry sets of good semigroups. This generalizes to any $d\ge 2$ the results of a recent paper of D'Anna, Guerrieri and Micale, which are proved in the case $d=2$ and only for the standard Apéry set with respect to the smallest nonzero element. Several new results describing good semigroups in ${\mathbb{N}}^d$ are also provided.

30 pages, 7 figures