Betti Numbers of the Tangent Cones of Monomial Space Curves

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