Tangent Cone of Numerical Semigroup Rings of Embedding Dimension Three

Abstract

Abstract. In this paper, we give new characterizations of the Buchsbaum and Cohen-Macaulay properties of the tangent cone gr m (R), where (R, m) is a numerical semigroup ring of embedding dimension 3. In particular, we confirm the conjectures raised by Sapko on the Buchsbaumness of gr m (R).

Goto Numbers of a Numerical Semigroup Ring and the Gorensteiness of Associated Graded Rings

“…It was shown by Heinzer and Swanson (2008, Theorem 5.10) that g(a 1 ) = τ if ν = 2. On the other hand, S =< 7, 11, 20 > and S =< 11, 14, 21 > are examples due to Shen (2008) of semigroups for which τ < g(a i ) for all i. Notice that the latter is symmetric and M-additive.…”

On pseudo symmetric monomial curves

“…In this section, we give conditions for the Cohen-Macaulayness of the tangent cone. For some recent and past activity about the tangent cone of C S , see [1,2,10,23,34,35].…”

The index of a numerical semigroup ring

“…Remark 3.1. If R is a Gorenstein numerical semigroup ring, then [9,Lemma 2.5] shows that N a1 = ord(f (H) + a 1 ) + 1. A direct computation, which we omit here, similar in difficulty and length to the proof of Theorem 2.2, shows that when edim(R) = 3 we have N ai = ord(f (H) + a j ) + 1 for all 1 ≤ j ≤ 3.…”