Algebra & Number Theory

Adjoint ideals and a correspondence between log canonicity and $F$-purity

Shunsuke Takagi

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Abstract

This paper presents three results on F-singularities. First, we give a new proof of Eisenstein’s restriction theorem for adjoint ideal sheaves using the theory of F-singularities. Second, we show that a conjecture of Mustaţă and Srinivas implies a conjectural correspondence of F-purity and log canonicity. Finally, we prove this correspondence when the defining equations of the variety are very general.

Article information

Source
Algebra Number Theory, Volume 7, Number 4 (2013), 917-942.

Dates
Received: 1 July 2011
Revised: 23 April 2012
Accepted: 27 May 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729986

Digital Object Identifier
doi:10.2140/ant.2013.7.917

Mathematical Reviews number (MathSciNet)
MR3095231

Zentralblatt MATH identifier
1305.14010

Subjects
Primary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]
Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14F18: Multiplier ideals

Keywords
adjoint ideals test ideals $F$-pure singularities log canonical singularities

Citation

Takagi, Shunsuke. Adjoint ideals and a correspondence between log canonicity and $F$-purity. Algebra Number Theory 7 (2013), no. 4, 917--942. doi:10.2140/ant.2013.7.917. https://projecteuclid.org/euclid.ant/1513729986


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References

  • M. Blickle, “Multiplier ideals and modules on toric varieties”, Math. Z. 248:1 (2004), 113–121.
  • L. Ein and M. Musta\commaaccenttă, “Jet schemes and singularities”, pp. 505–546 in Algebraic geometry. Part 2 (Seattle, 2005), edited by D. Abramovich et al., Proc. Sympos. Pure Math. 80, Amer. Math. Soc., Providence, RI, 2009.
  • E. Eisenstein, “Generalization of the restriction theorem for multiplier ideals”, preprint, 2010.
  • R. Fedder, “$F$-purity and rational singularity”, Trans. Amer. Math. Soc. 278:2 (1983), 461–480.
  • N. Hara, “A characterization of rational singularities in terms of injectivity of Frobenius maps”, Amer. J. Math. 120:5 (1998), 981–996.
  • N. Hara and K.-I. Watanabe, “F-regular and F-pure rings vs. log terminal and log canonical singularities”, J. Algebraic Geom. 11:2 (2002), 363–392.
  • N. Hara and K.-I. Yoshida, “A generalization of tight closure and multiplier ideals”, Trans. Amer. Math. Soc. 355:8 (2003), 3143–3174.
  • D. Hernández, “$F$-purity versus log canonicity for polynomials”, preprint, 2011.
  • H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II”, Ann. of Math. 79 (1964), 205–326.
  • M. Hochster and C. Huneke, “Tight closure in equal characteristic zero”, preprint, 1999, http://www.math.lsa.umich.edu/~hochster/tcz.ps.
  • J. A. Howald, “Multiplier ideals of monomial ideals”, Trans. Amer. Math. Soc. 353:7 (2001), 2665–2671.
  • J. Jang, “The ordinarity of an isotrivial elliptic fibration”, Manuscripta Math. 134:3-4 (2011), 343–358.
  • M. Kawakita, “Inversion of adjunction on log canonicity”, Invent. Math. 167:1 (2007), 129–133.
  • M. Kawakita, “On a comparison of minimal log discrepancies in terms of motivic integration”, J. Reine Angew. Math. 620 (2008), 55–65.
  • Y. Kawamata, “Crepant blowing-up of $3$-dimensional canonical singularities and its application to degenerations of surfaces”, Ann. of Math. $(2)$ 127:1 (1988), 93–163.
  • J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge University Press, 1998.
  • R. Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 49, Springer, Berlin, 2004.
  • V. B. Mehta and V. Srinivas, “Normal $F$-pure surface singularities”, J. Algebra 143:1 (1991), 130–143.
  • L. E. Miller and K. Schwede, “Semi-log canonical vs $F$-pure singularities”, J. Algebra 349 (2012), 150–164.
  • M. Musta\commaaccenttă and V. Srinivas, “Ordinary varieties and the comparison between multiplier ideals and test ideals”, Nagoya Math. J. 204 (2011), 125–157.
  • A. Ogus, “Hodge cycles and crystalline cohomology”, pp. 357–414 in Hodge cycles, motives, and Shimura varieties, edited by P. Deligne et al., Lecture Notes in Mathematics 900, Springer, Berlin, 1982.
  • K. Schwede, “Generalized test ideals, sharp $F$-purity, and sharp test elements”, Math. Res. Lett. 15:6 (2008), 1251–1261.
  • K. Schwede, “$F$-adjunction”, Algebra Number Theory 3:8 (2009), 907–950. http://msp.org/idx/mr/2011b:14006MR2011b:14006
  • K. Schwede, “Test ideals in non-$\mathbb{Q}$-Gorenstein rings”, Trans. Amer. Math. Soc. 363:11 (2011), 5925–5941.
  • K. Schwede and K. Tucker, “On the behavior of test ideals under finite morphisms”, preprint, 2012. To appear in J. Algebraic Geom.
  • J.-P. Serre, “Groupes de Lie $l$-adiques attachés aux courbes elliptiques”, pp. 239–256 in Les tendances géométriques en algébre et théorie des nombres, Centre National de la Recherche Scientifique, Paris, 1966.
  • T. Shibuta and S. Takagi, “Log canonical thresholds of binomial ideals”, Manuscripta Math. 130:1 (2009), 45–61.
  • S. Takagi, “F-singularities of pairs and inversion of adjunction of arbitrary codimension”, Invent. Math. 157:1 (2004), 123–146.
  • S. Takagi, “An interpretation of multiplier ideals via tight closure”, J. Algebraic Geom. 13:2 (2004), 393–415.
  • S. Takagi, “Formulas for multiplier ideals on singular varieties”, Amer. J. Math. 128:6 (2006), 1345–1362.
  • S. Takagi, “A characteristic $p$ analogue of plt singularities and adjoint ideals”, Math. Z. 259:2 (2008), 321–341.
  • S. Takagi, “Adjoint ideals along closed subvarieties of higher codimension”, J. Reine Angew. Math. 641 (2010), 145–162.
  • K. Watanabe, “Study of $F$-purity in dimension two”, pp. 791–800 in Algebraic geometry and commutative algebra, vol. II, edited by H. Hijikata et al., Kinokuniya, Tokyo, 1988.