Discreteness and rationality of $F$-jumping numbers on singular varieties

We prove that the $F$-jumping numbers of the test ideal $\tau(X; \Delta, \ba^t)$arediscreteandrationalundertheassumptionsthat$X$ is a normal and $F$-finite variety over a field of positive characteristic $p$, $K_X+\mathrm{\Delta }$ is $\text{bQ}$-Cartier of index not divisible $p$, and either $X$ is essentially of finite type over a field or the sheaf of ideals $\text{ba}$ 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.