Periodicity of betti numbers of monomial curves

Abstract

Let $K$ be an arbitrary field. Let $\a = (a_1< ... <a_n)$ be a sequence of positive integers. Let $C(\a)$ be the affine monomial curve in ${\mathbb{A}}^n$ parametrized by $t\to (t^{a_1},...,t^{a_n})$. Let $I(\a)$ be the defining ideal of $C(\a)$ in $K[x_1,...,x_n]$. For each positive integer $j$, let $\a+j$ be the sequence $(a_1+j,...,a_n+j)$. In this paper, we prove the conjecture of Herzog and Srinivasan saying that the betti numbers of $I(\a + j)$ are eventually periodic in $j$ with period $a_n-a_1$. When $j$ is large enough, we describe the betti table for the closure of $C(\a+j)$ in $\PP^n$.

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