Minimal binomial systems of generators for the ideals of certain monomial curves

Abstract

Let $a,b$ and $n>1$ be three positive integers such that $a$ and ${\sum }_{j=0}^{n-1}{b}^j$ are relatively prime. In this paper, we prove that the toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by $\{{\sum }_{j=0}^{n-1}{b}^j\}\cup \{{\sum }_{j=0}^{n-1}{b}^j+a{\sum }_{j=0}^{i-2}{b}^j\mid i=2,\dots ,n\}$ is determinantal. Moreover, we prove that for $n>3$, the ideal $I$ has a unique minimal system of generators if and only if $a<b-1$.

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