A note on the multiplier ideals of monomial ideals

Let a aS dagger a",[x (1), aEuro broken vertical bar, x (n) ] be a monomial ideal and J (a (c) ) the multiplier ideal of a with coefficient c. Then J (a (c) ) is also a monomial ideal of a",[x (1), aEuro broken vertical bar, x (n) ], and the equality J (a (c) ) = a implies that 0 < c < n + 1. We mainly discuss the problem when J (a) = a or for all 0 < epsilon < 1. It is proved that if J (a) = a then a is principal, and if holds for all 0 < epsilon < 1 then a = (x (1), aEuro broken vertical bar, x (n) ). One global result is also obtained. Let be the ideal sheaf on a"(TM) (n-1) associated with a. Then it is proved that the equality J () = implies that is principal.