Betti Numbers of the Tangent Cones of Monomial Space Curves

Abstract

Let \(H = \langle n_1, n_2,n_3\rangle \) be a numerical semigroup. Let \(\widetilde{H}\) be the interval completion of H, namely the semigroup generated by the interval \(\langle n_1,n_1+1, \ldots , n_3\rangle \). Let K be a field and K[H] the semigroup ring generated by H. Let \(I_H^{*}\) be the defining ideal of the tangent cone of K[H]. In this paper, we describe the defining equations of \(I_H^{*}\). From that, we prove the Herzog-Stamate conjecture for monomial space curves stating that \(\beta _i(I_H^{*}) \le \beta _i(I_{\widetilde{H}}^{*})\) for all i, where \(\beta _i(I_H^{*})\) and \(\beta _i(I_{\widetilde{H}}^{*})\) are the ith Betti numbers of \(I_H^{*}\) and \(I_{\widetilde{H}}^{*}\) respectively.