ON F-PURE THRESHOLDS
By S. Takagi, Kei-ichi Watanabe. 2003
139 citations recorded by the source corpus.
Using the Frobenius map, we introduce a new invariant for a pair (R,a) of a ring R and an ideal a⊂R, which we call the F-pure threshold c(a) of a, and study its properties. We see that the F-pure threshold characterizes several ring theoretic properties. By virtue of Hara and Yoshida's result [Trans. Amer. Math. Soc. 355 (2003) 3143], the F-pure threshold c(a) in characteristic zero corresponds to the log canonical threshold lc(a) which is an important invariant in birational geometry. Using the F-pure threshold, we prove some ring theoretic properties of three-dimensional terminal singularities of characteristic zero. Also, in fixed prime characteristic, we establish several properties of the F-pure threshold similar to those of the log canonical threshold with quite simple proofs.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/4e4607a03740c9c4824ecd0a264b8b6beeeb98a9
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Citations in the corpus
- p −1-Linear Maps in Algebra and Geometry (2012, background): “, at the origin m= hx1,...,xni, is defined as follows: lim e−→∞ max{l|fl∈/m[pe]} pe . Prove that this limit exists in general and then show that cm(x3−y2) = 5 6 if p= 7. See [MTW05] for solutions, cf. [TW04] . 7. Change of variety In this section, we describe how p−e-linear maps change under common c”
- IMPANGA lecture notes on log canonical thresholds (2011, background): “tisfies properties similar to those of the log canonical threshold, and it is related with this one in a subtle way via reduction mod p. The F-pure threshold has been introduced by Takagi and Watanabe [TW] when the ambient variety is fairly general. In what follows we will focus on the case of ambien”
- Hodge theory meets the minimal model program: a survey of log canonical and Du Bois singularities (2009, background): “re in [HH90], a precise correspondence between several classes of singularities was established. See, for example, [FW89], [MS91], [HW02], [Smi97], [Har98], [MS97], [Smi00], [Har05], [HY03], [Tak04], [TW04], [Tak08]. The second name author partially extended this correspondence in his doctoral disse”
- F-adjunction (2009, unknown): “In the case where R is a nonlocal ring, these definitions of strong F-regularity and sharp F-purity are slightly more general than the ones given in [Takagi 2004a; Takagi and Watanabe 2004; Schwede 2008a; 2008b].”
- A survey of test ideals (2011, background): “runs over all φ∈HomR(R1/p e,R). Let ǫ= 1 pN−1 , so that xa⌈t(pe−1)⌉ ⊆ a⌈(t+ǫ)(pe−1)⌉ for all 0 ≤e≤N. Thus, we have τ(R,at) ⊆ X e≥0 X φ φ((ca⌈(t+ǫ)(p e−1)⌉)1/p ) = τ(R,at+ǫ) as desired. Definition 6.7. [TW04], [MTW05], [BMS08] A positive real numberξis called an F-jumping number of the ideal a if τ(aξ”
- Discreteness and rationality of F-jumping numbers on singular varieties (2009, unknown): “et of F-jumping numbers, and in particular the first of these, called the F-pure threshold which is the analog of the log canonical threshold from characteristic zero, have been studied for example in [TW04], [MTW05], [HMTW08] and [Mus09]. Already in [MTW05] it was observed that the F-jumping numbers”
- F-thresholds and Bernstein–Sato Polynomials (2004, methodology): “deals in characteristic zero, it is not surprising that the F-thresholds behave in a similar way with the jumping coefficients of the multiplier ideals from [ELSV]. Such an analogy was also stressed in [TW], where it was shown that the smallest F-threshold c(a) behaves in the same way as the smallest ”
- ON THE BEHAVIOR OF TEST IDEALS UNDER FINITE MORPHISMS (2010, background): “e map is surjective. Given a strongly F-regular ring Rand a Weil divisor Bon X= SpecR, the F-pure threshold of (X,B) is defined to be the real number c(X,B) := sup{λ|(X,λB) is strongly F-regular}; see [TW04] and [MTW05]. It is a characteristic p>0 analog of the log canonical threshold. Corollary 7”
- Stability of valuations and Kollár components (2016, background): “tiplicity. The latter quantity was rst considered in the smooth case in [dFEM04], and since then its variants has been studied in many other work, including positive characteristic versions (see e.g. [TW04]). Its relevance to the normalized volume appeared in [Li15a, Example 2.7]. We have the follow”
- F-thresholds of hypersurfaces (2007, background): “en τ(aλ−pε ) = τ(aλ). As we have already mentioned, there are many analogies between the F-pure threshold and acharacteristic zero invariant that is very much studied, the logcanonical threshold (see [TW] and also [MTW]). However, in characteristic zero there is not much 18 M. Blickle, M. Musta¸ta a”
- Integral closure of ideals, rings, and modules (2006, background): “precise in his paper [288] dealing with singularities of pairs (see also [290]).”
- F-injective singularities are Du Bois (2008, background): “It is still an open question whether log canonical singularities are of dense F -pure type (although it is known in certain special cases; see [Har98b], [MS91], [Sri91], and [TW04]).”
- A Guided Tour to Normalized Volume (2018, background): “ally closed field k of positive characteristic. Let x ∈X be a closed point. We define the F-volume of (x ∈X) as Fvol(x,X) := inf a: mx-primary fpt(X;a)nmult(a). Similar to [dFEM03], Takagi and Watanabe [TW04] showed that if x ∈X is a smooth point, then Fvol(x,X) = nn. Another interesting invariant of ”