Aug 15, 2008 - Abstract. In this paper, we study the tangent cone of numerical semigroup ... suggest a positive answer to the following question. Question 1.1.
arXiv:0808.2162v1 [math.AC] 15 Aug 2008
TANGENT CONE OF NUMERICAL SEMIGROUP RINGS WITH SMALL EMBEDDING DIMENSION YIHUANG SHEN Abstract. In this paper, we study the tangent cone of numerical semigroup rings with small embedding dimension d. For d = 3, we give characterizations of the Buchsbaum and Cohen-Macaulay properties and for d = 4, we give a characterization of the Gorenstein property. In particular, when d = 4 and the tangent cone is Gorenstein, the initial form ideal of the defining ideal is 5-generated.
1. Introduction Throughout this paper we fix N = {1, 2, 3, · · · } and N0 = {0, 1, 2, · · · }. Recall that P a numerical semigroup G = hn1 , · · · , nd i generated by n1 , . . . , nd ∈ N is the n set { i=1 ai ni : ai ∈ N0 }. For simplicity, we always assume that G is minimally generated by these generators, with n1 < · · · < nd and gcd(n1 , . . . , nd ) = 1, unless stated otherwise. Let k be a field of characteristic 0, and t an indeterminate over k. Then as a subring of the power series ring V = k[[t]], R = k[[tn1 , . . . , tnd ]] is the numerical semigroup ring associated to G with m = (tn1 , . . . , tnd )R being the unique maximal ideal. R is the homomorphic image of the power series ring S = k[[x1 , . . . , xd ]] by mapping xi to tni . Let n be the unique maximal ideal of S, and I the kernel of this surjective map. I is a binomial ideal. We denote the kernel of the natural map grn (S) → grm (R) by I ∗ , and call it the initial form ideal of I. When the embedding dimension d = 3, Herzog [11] gave a complete characterization of the defining ideal I. In particular the minimal number of generators µ(I) ≤ 3. It is also proven by Robbiano and Valla [14] and Herzog [12] that the associated graded ring grm (R) is Cohen-Macaulay if and only if the initial form ideal I ∗ is generated by at most 3 elements. In this paper, we give another characterization for the Cohen-Macaulay property in terms of the index of nilpotency sQ (m) and the reduction number where Q = (tn1 )R is a principal reduction rQ (m), of m. Recall that sQ (m) = min s | ms+1 ⊆ Q . For d = 3, we also study the 0-th local cohomology module of the tangent cone. And we are able to characterize the k-Buchsbaum properties of grm (R) for k = 1, 2, in terms of the length of this local cohomology module. In particular, this answers the conjectures raised by Sapko [15]. When d = 4 and the numerical semigroup G is symmetric, Bresinsky [3] gave a complete description of the defining ideal I. In particular, it is well-known now that µ(I) ≤ 5. In this paper, we also study the initial form ideal of I. When the tangent cone grm (R) is Gorenstein, we show that I ∗ is also 5-generated. Meanwhile, Arslan 1991 Mathematics Subject Classification. Primary 13A30, Secondary 13P10, 13H10. 1