Markov complexity of monomial curves

Let $\mathcal{A}=\{{\bf a}_1,\ldots,{\bf a}_n\}\subset\Bbb{N}^m$. We give an algebraic characterization of the universal Markov basis of the toric ideal $I_{\mathcal{A}}$. We show that the Markov complexity of $\mathcal{A}=\{n_1,n_2,n_3\}$ is equal to two if $I_{\mathcal{A}}$ is complete intersection and equal to three otherwise, answering a question posed by Santos and Sturmfels. We prove that for any $r\geq 2$ there is a unique minimal Markov basis of $\mathcal{A}^{(r)}$. Moreover, we prove that for any integer $l$ there exist integers $n_1,n_2,n_3$ such that the Graver complexity of $\mathcal{A}$ is greater than $l$.

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