On the Cohen–Macaulayness of tangent cones of monomial curves inA4(K)

Abstract

In this paper we give necessary and sufficient conditions for the Cohen-Macaulayness of the tangent cone of a monomial curve in the 4-dimensional affine space. We study particularly the case where C is a Gorenstein noncomplete intersection monomial curve.

One-dimensional Gorenstein local rings with decreasing Hilbert function

“…Every case verifies the assumptions of Lemma 4.3.iii. Hence we can apply this result: note that in cases (1) . .…”

“…In Section 3 we prove the main result, see Theorem 3.3; moreover we give explicit examples of one-dimensional Gorenstein local semigroup rings with decreasing Hilbert function and other interesting examples based on the above constructions; see e.g. Example 3.4 with Hilbert function [1,53,54,54,53,53,56,59, 61, 63, 64 →], Example 3.9 with Hilbert function decreasing at many levels, and Example 3.10 for a ring with smaller multiplicity and embedding dimension. Finally the appendix contains the technical results needed to prove Theorem 2.9.…”

“…Example 3.7. Consider S := 30, 33, 37, 64, 68, 69, 71, 72, 73, 75, −→ 89, 91, 92, 95 that has Hilbert function [1,27,26,25,24,27,28,29,30, →] and set K := K(S) + 66 ⊆ S. Then the semigroup S ✶ 33 K is symmetric and has Hilbert function [1, 54, 55, 55, 54, 57, 58, 59, 60 →]. We also note that S is not almost symmetric by Proposition 1.4.…”

“…Let S be the semigroup of Construction 2.6 with ℓ = 15, that has 258 minimal generators. According to GAP [17], the symmetric semigroup S ✶ 957 K(S) has Hilbert function [1,514,514,513,512, 511, 510, 509, 508, 507, 506, 505, 504, 503, 502, 500, 523, H S (17), . .…”

“…• In [1] F. Arslan, A. Katsabekis, and M. Nalbandiyan, for other families of Gorenstein 4-generated numerical semigroup rings;…”