Hilbert series of tangent cones for Gorenstein monomial curves in A4(K)

Abstract

In this paper, we study the Hilbert series of the tangent cone of Gorenstein monomial curves in the 4dimensional affine space. We give an explicit formula for the reduced Hilbert series of the tangent cone of a noncomplete intersection Gorenstein monomial curve whose tangent cone is Cohen-Macaulay.

Minimal free resolutions of the tangent cones for Gorenstein monomial curves

“…Suppose that I S is given as in the Case 1(b) when α 3 > α 32 + α 34 . (i) By Proposition 2.7 in [9], if α 32 < α 42 and α 14 ≤ α 34 , then,…”

“…In [3], Arslan and Mete determined the common arithmetic conditions satisfied by the generators of the defining ideals of C S and under these conditions they found the generators of the tangent cone of C S . In [2], they provided necessary and sufficient conditions for the Cohen-Macaulayness of the tangent cone of C S in all six permutations and gave the following theorem: Recently, Katsabekis gave the remaining standard bases for I S in [9] using above conditions for the Cohen-Macaulayness of the tangent cone of C S .…”

“…(i) Assume that α 32 < α and α 14 ≤ α 34 . Then R/I Recently, Katsabekis gave the remaining standard bases for I S in [9] using above conditions for the Cohen-Macaulayness of the tangent cone of C S .…”

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

“…In [2], they provided necessary and sufficient conditions for the Cohen-Macaulayness of the tangent cone of C S in all six permutations and gave the following theorem: Recently, Katsabekis gave the remaining standard bases for I S in [9] using above conditions for the Cohen-Macaulayness of the tangent cone of C S .…”

“…ii) By Proposition 2.23 in[9], (1) if α 43 ≤ α 23 and α 24 < α 14 then {f 1 , f 2 , f 3 , f 4 , f 5 , f 6 = x α2+α12 if α 43 ≤ α 23 and α 14 ≤ α 24 then {f are standard bases for I S . Since I = π i (I S * ) which sends x 1 to 0, then the generators of the defining ideal of I is generated by G * = (x the case (i).…”