On the Periodicity of the First Betti Number of the Semigroup Rings Under Translations

Let k be a field of characteristic zero. Given an ordered 3-tuple of positive integers a = (a, b, c) and for j ∈ N≥1, a family of sequences a _j = ( j, a + j, a + b + j, a + b + c + j ), we consider the collection of monomial curves in A⁴ associated with aj . The Betti numbers of the Semigroup rings collection associated with a j are conjectured to be eventually periodic with period a + b + c by Herzog and Srinivasan. Let p ∈ N, in this paper, we prove that for a = (p(b+c), b, c) or a = (a, b, p(a+b)) in the collection of defining ideals associated with aj, for large j the ideals are complete intersections if and only if (a+b+c)| j .Moreover, the complete intersections are periodic with the conjectured period.