The jumping coefficients of non-QQ-Gorenstein multiplier ideals

Let a⊂OXa⊂OX be a coherent ideal sheaf on a normal complex variety X  , and let c≥0c≥0 be a real number. De Fernex and Hacon associated a multiplier ideal sheaf to the pair (X,ac)(X,ac) which coincides with the usual notion whenever the canonical divisor KXKX is QQ-Cartier. We investigate the properties of the jumping numbers associated to these multiplier ideals.We show that the set of jumping numbers of a pair is unbounded, countable and satisfies a certain periodicity property. We then prove that the jumping numbers form a discrete set of real numbers if the locus where KXKX fails to be QQ-Cartier is zero-dimensional. It follows that discreteness holds whenever X is a threefold with rational singularities.Furthermore, we show that the jumping numbers are rational and discrete if one removes from X   a closed subset W⊂XW⊂X of codimension at least three, which does not depend on aa. We also obtain that outside of W  , the multiplier ideal reduces to the test ideal modulo sufficiently large primes p≫0p≫0.