Minimal free resolutions of the tangent cones for Gorenstein monomial curves

Abstract

We study the minimal free resolution of the tangent cone of Gorenstein monomial curves in affine 4-space. We give the explicit minimal free resolution of the tangent cone of noncomplete intersection Gorenstein monomial curve whose tangent cone has five minimal generators and show that the possible Betti sequences are (1, 5, 6, 2) and (1, 5, 5, 1). Moreover, we compute the Hilbert function of the tangent cone of these families as a result.

On the Betti numbers of the tangent cones for Gorenstein monomial curves

“…Case 1(b) : Suppose that I S is minimally generated by the polynomials (1, 5, 5, 1) and (1, 5, 6, 2) , as given in [11]. Here, the other possible condition α 3 > α 32 + α 34 will be considered.…”

“…Under the restriction α 2 ≤ α 21 + α 24 by Theorem 2.2 and one possible condition α 3 ≤ α 32 + α 34 , the Betti sequences for the Cohen-Macaulay tangent cone of C S are (1, 5, 5, 1) and (1, 5, 6, 2) , as given in [11].…”

“…C S has Cohen-Macaulay tangent cone only under the conditions α 2 ≤ α 21 + α 23 and α 3 ≤ α 31 + α by [2]. Mete and Zengin [11] gave the explicit minimal free resolution for the tangent cone of C S and showed that the Betti sequence of its tangent cone is (1,5,6,2) .…”

4-generated pseudo symmetric monomial curves with not Cohen–Macaulay tangent cones

“…The conjecture is open for local rings corresponding to 4-generated symmetric semigroups with α 2 > α 21 + α 24 . For some recent work on the monotonicity of the Hilbert function, see [1][2][3][11][12][13].…”

Betti Numbers for Certain Cohen–macaulay Tangent Cones

“…Since homogeneous-type semigroups have Cohen-Macaulay tangent cones, we restrict our attention to monomial curves having Cohen-Macaulay tangent cones in this article. The problem of determining the Betti sequence for the tangent cone (see [12, Problem 9.9]) was studied for 4-generated symmetric monomial curves by Mete and Zengin [10]. In this paper, we focus on the next interesting case of 4-generated pseudo-symmetric monomial curves.…”