Stability of test ideals of divisors with small multiplicity

Let (X, Δ) be a log pair in characteristic p> 0 and P be a (not necessarily closed) point of X. We show that there exists a constant δ> 0 such that the test ideal τ(X, Δ) , a characteristic p analogue of a multiplier ideal, does not change at P under the perturbation of Δ by any R-divisor with multiplicity less than δ. As an application, we prove that if D is an R-Cartier R-divisor on a strongly F-regular projective variety, then the non-nef locus of D coincides with the restricted base locus of D. This is a generalization of a result of Mustaţǎ to the singular case and can be viewed as a characteristic p analogue of a result of Cacciola–Di Biagio.