On the symbolic powers of defining ideals of monomial curves associated to generalized arithmetic sequences

Abstract

Let s , a , n , d be positive integers such that n ≥ 2 and GCD ( a , d ) = 1 . Let P denote the defining ideal of the monomial curve associated to the sequence a , sa + d , … , sa + nd . In this survey, we investigate the symbolic powers of P. We first show that for each i > 1 the equality P ( i ) = P i holds if and only if either n = 2 and a even or n = 3 , i = 2 and a ≡ 2 ( mod 3 ) . The finitely generated property of the symbolic Rees algebra R S ( P ) are also explored.