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 $\tilde H$ be the interval completion of $H$, namely the semigroup generated by the interval $\langle n_1, n_1+1, \ldots, n_3\rangle$. 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 establish the Herzog-Stamate conjecture for monomial space curves stating that $\beta_i(I_H^*) \le \beta_i(I_{\tilde H}^*)$ for all $i$, where $\beta_i(I_H^*)$ and $\beta_i(I_{\tilde H}^*)$ are the $i$th Betti numbers of $I_H^*$ and $I_{\tilde H}^*$ respectively.