## Algebra & Number Theory

### Log canonical thresholds, $F$-pure thresholds, and nonstandard extensions

#### Abstract

We present a new relation between an invariant of singularities in characteristic zero (the log canonical threshold) and an invariant of singularities defined via the Frobenius morphism in positive characteristic (the $F$-pure threshold). We show that the set of limit points of sequences of the form $(cp)$, where $cp$ is the $F$-pure threshold of an ideal on an $n$-dimensional smooth variety in characteristic $p$, coincides with the set of log canonical thresholds of ideals on $n$-dimensional smooth varieties in characteristic zero. We prove this by combining results of Hara and Yoshida with nonstandard constructions.

#### Article information

Source
Algebra Number Theory, Volume 6, Number 7 (2012), 1459-1482.

Dates
Received: 1 June 2011
Revised: 16 November 2011
Accepted: 20 December 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729891

Digital Object Identifier
doi:10.2140/ant.2012.6.1459

Mathematical Reviews number (MathSciNet)
MR3007155

Zentralblatt MATH identifier
1262.13006

#### Citation

Bhatt, Bhargav; Hernández, Daniel; Miller, Lance Edward; Mustaţă, Mircea. Log canonical thresholds, $F$-pure thresholds, and nonstandard extensions. Algebra Number Theory 6 (2012), no. 7, 1459--1482. doi:10.2140/ant.2012.6.1459. https://projecteuclid.org/euclid.ant/1513729891

#### References

• M. Blickle, M. Musta\commaaccenttă, and K. E. Smith, “Discreteness and rationality of $F$-thresholds”, Michigan Math. J. 57 (2008), 43–61.
• M. Blickle, M. Musta\commaaccenttă, and K. E. Smith, “$F$-thresholds of hypersurfaces”, Trans. Amer. Math. Soc. 361:12 (2009), 6549–6565.
• M. Blickle, K. Schwede, S. Takagi, and W. Zhang, “Discreteness and rationality of $F$-jumping numbers on singular varieties”, Math. Ann. 347:4 (2010), 917–949.
• P. Deligne and L. Illusie, “Relèvements modulo $p\sp 2$ et décomposition du complexe de de Rham”, Invent. Math. 89:2 (1987), 247–270.
• L. van den Dries and K. Schmidt, “Bounds in the theory of polynomial rings over fields: A nonstandard approach”, Invent. Math. 76:1 (1984), 77–91.
• L. Ein and M. Musta\commaaccenttă, “Invariants of singularities of pairs”, pp. 583–602 in International Congress of Mathematicians (Madrid, 2006), vol. 2, edited by M. Sanz-Solé et al., European Mathematical Society, Zürich, 2006.
• T. de Fernex and M. Musta\commaaccenttă, “Limits of log canonical thresholds”, Ann. Sci. Éc. Norm. Supér. $(4)$ 42:3 (2009), 491–515.
• R. Goldblatt, Lectures on the hyperreals: An introduction to nonstandard analysis, Graduate Texts in Mathematics 188, Springer, New York, 1998.
• 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.
• J. Kollár, “Singularities of pairs”, pp. 221–287 in Algebraic geometry, Part 1 (Santa Cruz, CA, 1995), edited by J. Kollár et al., Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI, 1997.
• R. Lazarsfeld, Positivity in algebraic geometry, II: Positivity for vector bundles, and multiplier ideals, Ergeb. Math. Grenzgeb. (3) 49, Springer, Berlin, 2004.
• V. B. Mehta and V. Srinivas, “A characterization of rational singularities”, Asian J. Math. 1:2 (1997), 249–271.
• M. Musta\commaaccenttă, S. Takagi, and K.-i. Watanabe, “F-thresholds and Bernstein–Sato polynomials”, pp. 341–364 in European Congress of Mathematics (Stockholm, 2004), edited by A. Laptev, European Mathematical Society, Zürich, 2005.
• H. Schoutens, “Log-terminal singularities and vanishing theorems via non-standard tight closure”, J. Algebraic Geom. 14:2 (2005), 357–390.
• H. Schoutens, The use of ultraproducts in commutative algebra, Lecture Notes in Mathematics 1999, Springer, Berlin, 2010.
• S. Takagi and K.-i. Watanabe, “On F-pure thresholds”, J. Algebra 282:1 (2004), 278–297.