Betti numbers of the tangent cones of monomial space curves

Abstract

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