Abstract
Let \((X, \varDelta )\) 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 \(\delta >0\) such that the test ideal \(\tau (X, \varDelta )\), a characteristic p analogue of a multiplier ideal, does not change at P under the perturbation of \(\varDelta \) by any \(\mathbb {R}\)-divisor with multiplicity less than \(\delta \). As an application, we prove that if D is an \(\mathbb {R}\)-Cartier \(\mathbb {R}\)-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.
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Acknowledgements
The author wishes to express his gratitude to his supervisor Professor Shunsuke Takagi for his encouragement, valuable advice and suggestions. He is grateful to Doctor Sho Ejiri for his encouragement. He is also grateful to an anonymous referee for many useful suggestions and for pointing out many typos. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan.
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Sato, K. Stability of test ideals of divisors with small multiplicity. Math. Z. 288, 783–802 (2018). https://doi.org/10.1007/s00209-017-1913-0
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DOI: https://doi.org/10.1007/s00209-017-1913-0