Periodic Occurance of Complete Intersection Monomial Curves

Abstract

We study the complete intersection property of monomial curves in the family $\Gamma_{\aa + \jj} = {(t^{a_0 + j}, t^{a_1+j},..., t^{a_n + j}) ~ | ~ j \geq 0, ~ a_0 < a_1 <...< a_n}$. We prove that if $\Gamma_{\aa+\jj}$ is a complete intersection for $j \gg0$, then $\Gamma_{\aa+\jj+\underline{a_n}}$ is a complete intersection for $j \gg 0$. This proves a conjecture of Herzog and Srinivasan on eventual periodicity of Betti numbers of semigroup rings under translations for complete intersections. We also show that if $\Gamma_{\aa+\jj}$ is a complete intersection for $j \gg 0$, then $\Gamma_{\aa}$ is a complete intersection. We also characterize the complete intersection property of this family when $n = 3$.