Periodic occurrence of complete intersection monomial curves

We study the complete intersection property of monomial curves in the family. We prove that if is a complete intersection for j ≫ 0, then Γa+j+an is a complete intersection for j ≫ 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 is a complete intersection for j ≫ 0, then is a complete intersection. We also characterize the complete intersection property of this family when n = 3. © 2013 American Mathematical Society.