The Michigan Mathematical Journal

Rational singularities associated to pairs

Karl Schwede and Shunsuke Takagi

Source: Michigan Math. J. Volume 57 (2008), 625-658.

Primary Subjects: 14B05, 13A35
Secondary Subjects: 14J17

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.mmj/1220879429
Digital Object Identifier: doi:10.1307/mmj/1220879429
Mathematical Reviews number (MathSciNet): MR2492473
Zentralblatt MATH identifier: 05604552

References

I. M. Aberbach and C. Huneke, $F$-rational rings and the integral closures of ideals, Michigan Math. J. 49 (2001), 3--11.
Mathematical Reviews (MathSciNet): MR1827071
Digital Object Identifier: doi:10.1307/mmj/1008719031
Project Euclid: euclid.mmj/1008719031
M. Blickle, Multiplier ideals and modules on toric varieties, Math. Z. 248 (2004), 113--121.
Mathematical Reviews (MathSciNet): MR2092724
Digital Object Identifier: doi:10.1007/s00209-004-0655-y
J.-F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 65--68.
Mathematical Reviews (MathSciNet): MR877006
Digital Object Identifier: doi:10.1007/BF01405091
P. Du Bois, Complexe de de Rham filtré d'une variété singulière, Bull. Soc. Math. France 109 (1981), 41--81.
Mathematical Reviews (MathSciNet): MR613848
L. Ein, Multiplier ideals, vanishing theorems and applications, Algebraic geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math., 62, pp. 203--219, Amer. Math. Soc., Providence, RI, 1997.
Mathematical Reviews (MathSciNet): MR1492524
Zentralblatt MATH: 0978.14004
R. Elkik, Singularités rationnelles et déformations, Invent. Math. 47 (1978), 139--147.
Mathematical Reviews (MathSciNet): MR501926
Digital Object Identifier: doi:10.1007/BF01578068
------, Rationalité des singularités canoniques, Invent. Math. 64 (1981), 1--6.
Mathematical Reviews (MathSciNet): MR621766
Digital Object Identifier: doi:10.1007/BF01393930
R. Fedder, $F$-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461--480.
Mathematical Reviews (MathSciNet): MR701505
Digital Object Identifier: doi:10.2307/1999165
R. Fedder and K. Watanabe, A characterization of $F$-regularity in terms of $F$-purity, Commutative algebra (Berkeley, 1987), Math. Sci. Res. Inst. Publ., 15, pp. 227--245, Springer-Verlag, New York, 1989.
Mathematical Reviews (MathSciNet): MR1015520
Zentralblatt MATH: 0738.13004
H. Grauert and O. Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263--292.
Mathematical Reviews (MathSciNet): MR302938
Digital Object Identifier: doi:10.1007/BF01403182
N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981--996.
Mathematical Reviews (MathSciNet): MR1646049
Zentralblatt MATH: 0942.13006
Digital Object Identifier: doi:10.1353/ajm.1998.0037
------, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 1885--1906.
Mathematical Reviews (MathSciNet): MR1813597
Digital Object Identifier: doi:10.1090/S0002-9947-01-02695-2
N. Hara and S. Takagi, On a generalization of test ideals, Nagoya Math. J. 175 (2004), 59--74.
Mathematical Reviews (MathSciNet): MR2085311
Project Euclid: euclid.nmj/1114632095
N. Hara and K.-i. Watanabe, $F$-regular and $F$-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom. 11 (2002), 363--392.
Mathematical Reviews (MathSciNet): MR1874118
N. Hara, K.-i. Watanabe, and K.-I. Yoshida, $F$-rationality of Rees algebras, J. Algebra 247 (2002), 153--190.
Mathematical Reviews (MathSciNet): MR1873388
Digital Object Identifier: doi:10.1006/jabr.2001.8998
N. Hara and K.-I. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), 3143--3174.
Mathematical Reviews (MathSciNet): MR1974679
Digital Object Identifier: doi:10.1090/S0002-9947-03-03285-9
R. Hartshorne, Residues and duality, Lecture Notes in Math., 20, Springer-Verlag, Berlin, 1966.
Mathematical Reviews (MathSciNet): MR222093
Zentralblatt MATH: 0212.26101
M. Hochster and C. Huneke, Tight closure and strong $F$-regularity, Colloque en l'honneur de Pierre Samuel (Orsay, 1987) Mém. Soc. Math. France (N.S.) 38 (1989), 119--133.
Mathematical Reviews (MathSciNet): MR1044348
------, Tight closure, invariant theory, and the Briançon--Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31--116.
Mathematical Reviews (MathSciNet): MR1017784
Digital Object Identifier: doi:10.2307/1990984
------, Phantom homology, Mem. Amer. Math. Soc. 103 (1993).
Mathematical Reviews (MathSciNet): MR1144758
M. Hochster and J. L. Roberts, The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117--172.
Mathematical Reviews (MathSciNet): MR417172
Digital Object Identifier: doi:10.1016/0001-8708(76)90073-6
C. Huneke, Tight closure and its applications (with an appendix by Melvin Hochster), CBMS Reg. Conf. Ser. Math., 88, Amer. Math. Soc., Providence, RI, 1996.
Mathematical Reviews (MathSciNet): MR1377268
Zentralblatt MATH: 0930.13004
E. Hyry and K. E. Smith, On a non-vanishing conjecture of Kawamata and the core of an ideal, Amer. J. Math. 125 (2003), 1349--1410.
Mathematical Reviews (MathSciNet): MR2018664
Zentralblatt MATH: 1089.13003
Digital Object Identifier: doi:10.1353/ajm.2003.0041
E. Hyry and O. Villamayor, A Briançon--Skoda theorem for isolated singularities, J. Algebra 204 (1998), 656--665.
Mathematical Reviews (MathSciNet): MR1624420
Digital Object Identifier: doi:10.1006/jabr.1997.7386
M. Katzman, G. Lyubeznik, and W. Zhang, On the discreteness and rationality of jumping coefficients, preprint, arXiv:0706.3028.
M. Kawakita, Inversion of adjunction on log canonicity, Invent. Math. 167 (2007), 129--133.
Mathematical Reviews (MathSciNet): MR2264806
Digital Object Identifier: doi:10.1007/s00222-006-0008-z
Y. Kawamata, A generalization of Kodaira--Ramanujam's vanishing theorem, Math. Ann. 261 (1982), 43--46.
Mathematical Reviews (MathSciNet): MR675204
Digital Object Identifier: doi:10.1007/BF01456407
G. R. Kempf, Some quotient varieties have rational singularities, Michigan Math. J. 24 (1977), 347--352.
Mathematical Reviews (MathSciNet): MR491675
Digital Object Identifier: doi:10.1307/mmj/1029001952
Project Euclid: euclid.mmj/1029001952
G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Math., 339, Springer-Verlag, Berlin, 1973.
Mathematical Reviews (MathSciNet): MR335518
Zentralblatt MATH: 0271.14017
J. Kollár, Shafarevich maps and automorphic forms, Princeton Univ. Press, Princeton, NJ, 1995.
Mathematical Reviews (MathSciNet): MR1341589
------, Singularities of pairs, Algebraic geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math., 62, pp. 221--287, Amer. Math. Soc., Providence, RI, 1997.
Mathematical Reviews (MathSciNet): MR1492525
Zentralblatt MATH: 0905.14002
J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math., 134, Cambridge Univ. Press, Cambridge, 1998.
Mathematical Reviews (MathSciNet): MR1658959
J. Kollár et al., Flips and abundance for algebraic threefolds, Papers from the second summer seminar on algebraic geometry (Salt Lake City, 1991), Astérisque 211 (1992).
S. J. Kovács, Rational, log canonical, Du Bois singularities: On the conjectures of Kollár and Steenbrink, Compositio Math. 118 (1999), 123--133.
Mathematical Reviews (MathSciNet): MR1713307
Digital Object Identifier: doi:10.1023/A:1001120909269
------, A characterization of rational singularities, Duke Math. J. 102 (2000), 187--191.
Mathematical Reviews (MathSciNet): MR1749436
Digital Object Identifier: doi:10.1215/S0012-7094-00-10221-9
Project Euclid: euclid.dmj/1092749293
R. Lazarsfeld, Positivity in algebraic geometry. II, Ergeb. Math. Grenzgeb. (3), 49, Springer-Verlag, Berlin, 2004.
Mathematical Reviews (MathSciNet): MR2095472
J. Lipman and B. Teissier, Pseudorational local rings and a theorem of Briançon--Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), 97--116.
Mathematical Reviews (MathSciNet): MR600418
Digital Object Identifier: doi:10.1307/mmj/1029002461
Project Euclid: euclid.mmj/1029002461
G. Lyubeznik, $F$-modules: Applications to local cohomology and $D$-modules in characteristic $p>0,$ J. Reine Angew. Math. 491 (1997), 65--130.
Mathematical Reviews (MathSciNet): MR1476089
V. B. Mehta and V. Srinivas, A characterization of rational singularities, Asian J. Math. 1 (1997), 249--271.
Mathematical Reviews (MathSciNet): MR1491985
M. Mustaţ\v a, S. Takagi, and K.-i. Watanabe, $F$-thresholds and Bernstein--Sato polynomials, European congress of mathematics, pp. 341--364, European Mathematical Society, Zürich, 2005.
Mathematical Reviews (MathSciNet): MR2185754
K. Schwede, A simple characterization of Du Bois singularities, Compositio Math. 143 (2007), 813--828.
Mathematical Reviews (MathSciNet): MR2339829
K. Schwede, S. Kovács, and K. E. Smith, On a conjecture of Kollár that log canonical singularities are Du Bois,, preprint.
V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 105--203.
Mathematical Reviews (MathSciNet): MR1162635
A. K. Singh, Cyclic covers of rings with rational singularities, Trans. Amer. Math. Soc. 355 (2003), 1009--1024.
Mathematical Reviews (MathSciNet): MR1938743
Digital Object Identifier: doi:10.1090/S0002-9947-02-03186-0
K. E. Smith, Test ideals in local rings, Trans. Amer. Math. Soc. 347 (1995), 3453--3472.
Mathematical Reviews (MathSciNet): MR1311917
Digital Object Identifier: doi:10.2307/2155019
------, $F$-rational rings have rational singularities, Amer. J. Math. 119 (1997), 159--180.
Mathematical Reviews (MathSciNet): MR1428062
Zentralblatt MATH: 0910.13004
Digital Object Identifier: doi:10.1353/ajm.1997.0007
S. Takagi, $F$-singularities of pairs and inversion of adjunction of arbitrary codimension, Invent. Math. 157 (2004), 123--146.
Mathematical Reviews (MathSciNet): MR2135186
Digital Object Identifier: doi:10.1007/s00222-003-0350-3
------, A characteristic p analogue of plt singularities and adjoint ideals, Math. Z. (to appear).
Mathematical Reviews (MathSciNet): MR2390084
Digital Object Identifier: doi:10.1007/s00209-007-0227-z
S. Takagi and K.-i. Watanabe, On $F$-pure thresholds, J. Algebra 282 (2004), 278--297.
Mathematical Reviews (MathSciNet): MR2097584
Digital Object Identifier: doi:10.1016/j.jalgebra.2004.07.011
S. Takagi and K.-I. Yoshida, Generalized test ideals and symbolic powers, Michigan Math. J. (to appear).
J. D. Vélez, Openness of the $F$-rational locus and smooth base change, J. Algebra 172 (1995), 425--453.
Mathematical Reviews (MathSciNet): MR1322412
Digital Object Identifier: doi:10.1016/S0021-8693(05)80010-9
E. Viehweg, Vanishing theorems, J. Reine Angew. Math. 335 (1982), 1--8.
Mathematical Reviews (MathSciNet): MR667459
K.-i. Watanabe, $F$-rationality of certain Rees algebras and counterexamples to ``Boutot's theorem'' for $F$-rational rings, J. Pure Appl. Algebra 122 (1997), 323--328.
Mathematical Reviews (MathSciNet): MR1481095
Digital Object Identifier: doi:10.1016/S0022-4049(97)00064-9

2009 © The University of Michigan