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Betti Numbers of the Tangent Cones of Monomial Space Curves

Nguyen P. H. Lan , Nguyen Chanh Tu , $icon-email$ Thanh Vu

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 $I_{H}^{*}$ 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 $β_{i} (I_{H}^{*}) \leq β_{i} (I_{\tilde{H}}^{*})$ for all $i$ , where $β_{i} (I_{H}^{*})$ and $β_{i} (I_{\tilde{H}}^{*})$ are the $i$ th Betti numbers of $I_{H}^{*}$ and $I_{\tilde{H}}^{*}$ respectively.

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Betti Numbers of the Tangent Cones of Monomial Space Curves

Abstract