Ordinary varieties and the comparison between multiplier ideals and test ideals II

Author:
Mircea Mustaţă

Journal:
Proc. Amer. Math. Soc. **140** (2012), 805-810

MSC (2010):
Primary 13A35; Secondary 14F18, 14F30

Published electronically:
August 29, 2011

MathSciNet review:
2869065

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the following conjecture: if is a smooth -dimensional projective variety in characteristic zero, then there is a dense set of reductions to positive characteristic such that the action of the Frobenius morphism on is bijective. We also consider the conjecture relating the multiplier ideals of an ideal on a nonsingular variety in characteristic zero, and the test ideals of the reductions of to positive characteristic. We prove that the latter conjecture implies the former one.

**[BMS1]**Manuel Blickle, Mircea Mustaţă, and Karen E. Smith,*𝐹-thresholds of hypersurfaces*, Trans. Amer. Math. Soc.**361**(2009), no. 12, 6549–6565. MR**2538604**, 10.1090/S0002-9947-09-04719-9**[BMS2]**Manuel Blickle, Mircea Mustaţǎ, and Karen E. Smith,*Discreteness and rationality of 𝐹-thresholds*, Michigan Math. J.**57**(2008), 43–61. Special volume in honor of Melvin Hochster. MR**2492440**, 10.1307/mmj/1220879396**[BK]**Spencer Bloch and Kazuya Kato,*𝑝-adic étale cohomology*, Inst. Hautes Études Sci. Publ. Math.**63**(1986), 107–152. MR**849653****[ERT]**David Eisenbud, Alyson Reeves, and Burt Totaro,*Initial ideals, Veronese subrings, and rates of algebras*, Adv. Math.**109**(1994), no. 2, 168–187. MR**1304751**, 10.1006/aima.1994.1085**[Fe]**Richard Fedder,*𝐹-purity and rational singularity*, Trans. Amer. Math. Soc.**278**(1983), no. 2, 461–480. MR**701505**, 10.1090/S0002-9947-1983-0701505-0**[HY]**Nobuo Hara and Ken-Ichi Yoshida,*A generalization of tight closure and multiplier ideals*, Trans. Amer. Math. Soc.**355**(2003), no. 8, 3143–3174 (electronic). MR**1974679**, 10.1090/S0002-9947-03-03285-9**[Laz]**Robert Lazarsfeld,*Positivity in algebraic geometry. II*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR**2095472****[MS]**M. Mustaţă and V. Srinivas, Ordinary varieties and the comparison between multiplier ideals and test ideals, arXiv:1012.2818, to appear in Nagoya Math. J.**[MTW]**Mircea Mustaţǎ, Shunsuke Takagi, and Kei-ichi Watanabe,*F-thresholds and Bernstein-Sato polynomials*, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 341–364. MR**2185754****[MY]**Mircea Mustaţă and Ken-Ichi Yoshida,*Test ideals vs. multiplier ideals*, Nagoya Math. J.**193**(2009), 111–128. MR**2502910**

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Additional Information

**Mircea Mustaţă**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Email:
mmustata@umich.edu

DOI:
https://doi.org/10.1090/S0002-9939-2011-11240-1

Keywords:
Test ideals,
multiplier ideals,
ordinary variety

Received by editor(s):
December 18, 2010

Published electronically:
August 29, 2011

Additional Notes:
The author was partially supported by NSF grant DMS-0758454 and a Packard Fellowship.

Communicated by:
Irena Peeva

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.