We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numerical semigroups with Cohen-Macaulay tangent cones are of homogeneous type. In embedding dimension three, we classify all numerical semigroups of homogeneous type into numerical semigroups with complete intersection tangent cones and the homogeneous ones which are not symmetric with Cohen-Macaulay tangent cones. We also study the behavior of the homogeneous property by gluing and shiftings to construct large families of homogeneous numerical semigroups with Cohen-Macaulay tangent cones. In particular we show that these properties fulfill asymptotically in the shifting classes. Several explicit examples are provided along the paper to illustrate the property.

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We study arithmetic properties of tangent cones associated to affine monomial curves, using the concept of gluing. In particular we characterize the Cohen-Macaulay and Gorenstein properties of tangent cones of some families of monomial curves obtained by gluing. Moreover, we provide new families of monomial curves with non-decreasing Hilbert functions. introductionA monomial curve C in the affine space A d k over a field k consists on the set of points defined parametrically by X 1 = t m1 , . . . , X d = t m d , for some positive integers m 1 < · · · < m d . In order to be sure that different parameterizations give rise to different monomial curves, we may assume that gcd(m 1 , . . . , m d ) = 1.In fact, it is known that the set C is an affine variety whose coordinate ring issee for instance E. Reyes, R. H. Villarreal and L. Zárate [17]. The set S = {r 1 m 1 + · · · + r d m d ; r i ≥ 0} is a subset of the non-negative integers N ∪ {0} which is closed under addition, and the condition gcd(m 1 , . . . , m d ) = 1 is equivalent to the property # N \ S < ∞. In other words, S =< m 1 , . . . , m d > is a numerical semigroup minimally generated by the unique minimal system of generators {m 1 , . . . , m d }. The coordinate ring R is called the numerical semigroup ring associated to S and it is denoted by k[S]. Since we are interested in the arithmetical properties at the origin, which is the only singular point of the curve C, we will consider the ring R = k[[t m1 , . . . , t m d ]] = k[[S]]. Note that R is a complete one-dimensional local domain with maximal ideal m = (t m1 , . . . , t m d ).We also consider the tangent cone associated to k[[S]]; that is the graded ring G(S) := n≥0 m n /m n+1 .

For a given a local ring (A, m), we study the fiber cone of ideals in A with analytic spread one. In this case, the fiber cone has a structure as a module over its Noether normalization which is a polynomial ring in one variable over the residue field. One may then apply the structure theorem for modules over a principal domain to get a complete description of the fiber cone as a module. We analyze this structure in order to study and characterize in terms of the ideal itself the arithmetical properties and other numerical invariants of the fiber cone as multiplicity, reduction number or Castelnuovo-Mumford regularity.

In this paper we describe the structure of the tangent cone of a numerical semigroup ringwith multiplicity e (as a module over the Noether normalization determined by the fiber cone of the ideal generated by t e ) in terms of some classical invariants of the corresponding numerical semigroup. Explicit computations are also made by using the GAP system.