Numerical semigroups with maximal embedding dimension

Even though the study and relevance of maximal embedding dimension numerical semigroups arises in a natural way among the other numerical semigroups, they have become specially renowned due to the existing applications to commutative algebra via their associated semigroup ring (see for instance [1, 5, 15, 16, 99, 100]). They are a source of examples of commutative rings with some maximal properties. As we mentioned in the introduction of Chapter 1, this is partially due to the fact that the study of some attributes of an analytically unramified one-dimensional local domains can be performed via their value semigroups. Of particular interest are two subclasses of maximal embedding dimension numerical semigroups, which are those semigroups having the Arf property and saturated numerical semigroups. These two families are related with the problem of resolution of singularities in a curve. Inspired by [3], Lipman in [47] introduces and motivates the study of Arf rings. The characterization of these rings via their value semigroups yields the Arf property for numerical semigroups. The reader can find in [5] a considerable amount of characterizations of this property for numerical semigroups. Arf numerical semigroups have gained lately a particular interest due to their applications to algebraic error correcting codes (see [18, 7] and the references given therein). Saturated rings were introduced in three different ways by Zariski ([109]), PhamTeissier ([50]) and Campillo ([17]), though their definitions coincide for algebraically closed fields of zero characteristic. As for the Arf property, saturated numerical semigroups come into the scene after a characterization of saturated rings in terms of their value semigroups (see [26, 49]).