The jumping coefficients of non-[formula omitted]-Gorenstein multiplier ideals

Abstract

Let a⊂OX be a coherent ideal sheaf on a normal complex variety X, and let c≥0 be a real number. De Fernex and Hacon associated a multiplier ideal sheaf to the pair (X,ac) which coincides with the usual notion whenever the canonical divisor KX is Q-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 KX fails to be Q-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⊂X of codimension at least three, which does not depend on a. We also obtain that outside of W, the multiplier ideal reduces to the test ideal modulo sufficiently large primes p≫0.