Periodic occurrence of complete intersection monomial curves

Abstract:

We study the complete intersection property of monomial curves in the family $\Gamma _{\underline {a} + \underline {j}} = \{(t^{a_0 + j}, t^{a_1+j}, \ldots , t^{a_n + j}) ~ | ~ j \geq 0, ~ a_0 < a_1 < \cdots < a_n \}$. We prove that if $\Gamma _{\underline {a}+\underline {j}}$ is a complete intersection for $j \gg 0$, then $\Gamma _{\underline {a}+\underline {j}+{\underline {a}_n}}$ 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 $\Gamma _{\underline {a}+\underline {j}}$ is a complete intersection for $j \gg 0$, then $\Gamma _{\underline {a}}$ is a complete intersection. We also characterize the complete intersection property of this family when $n = 3$.