Discreteness and rationality of F-jumping numbers on singular varieties

We prove that the F-jumping numbers of the test ideal t(X; D, mathfraka^t){tau(X; Delta, mathfrak{a}^t)} are discrete and rational under the assumptions that X is a normal and F-finite scheme over a field of positive characteristic p, K _X  + Δ is mathbb Q{mathbb {Q}}-Cartier of index not divisible p, and either X is essentially of finite type over a field or the sheaf of ideals mathfraka{mathfrak{a}} is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero.