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Betti Numbers of the Tangent Cones of Monomial Space Curves
Nguyen P. H. Lan
,
Nguyen Chanh Tu
,
Thanh Vu
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 $i$th Betti numbers of $I_H^{*}$ and $I_{\widetilde{H}}^*$ respectively.