F-singularities of pairs and Inversion of Adjunction of arbitrary codimension

By S. Takagi. 2003

57 citations recorded by the source corpus.

We generalize the notions of F-regular and F-pure rings to pairs $(R,\mathfrak{a}^t)$ of rings R and ideals $\mathfrak{a} \subset{R}$ with real exponent t>0, and investigate these properties. These “F-singularities of pairs” correspond to singularities of pairs of arbitrary codimension in birational geometry. Via this correspondence, we prove a sort of Inversion of Adjunction of arbitrary codimension, which states that for a pair (X,Y) of a smooth variety X and a closed subscheme $Y\subsetneq{X}$, if the restriction (Z,Y|Z) to a normal ℚ-Gorenstein closed subvariety $Z\subsetneq{X}$ is klt (resp. lc), then the pair (X,Y+Z) is plt (resp. lc) near Z.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b32b5b8b3727147f93c4e2caa19ad482d353c370

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

ON F-PURE THRESHOLDS (2003, background): “However, it immediately follows from [ Ta , Theorem”
A survey of test ideals (2011, background): “s introduced in [Sch08]. This variant uses the ⌈(pe− 1)∆⌉or ⌈t(pe −1)⌉-exponent instead of ⌊t(pe −1)⌋or ⌈tpe⌉. One aspect that distinguishes it from previous F-purity deﬁnitions for pairs, see [HW02] [Tak04a], is that the test ideal of a sharply F-pure pair is always a radical ideal. See [Sch08, Cor”
F-thresholds and Bernstein–Sato Polynomials (2004, methodology): “f those t ∈ Q+ such that the pair (R,at) is F-pure. Under this extra assumption on F, since R is regular, the pair (R,at) is F-pure if and only if for q ≫ 0 we have a⌊t(q−1)⌋ ⊆ m[q] (see Lemma 3.9 in [Ta]). Here we use the notation ⌊α⌋ for the largest integer ≤ α. The above condition is equivalent w”
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].”
Globally F-regular and log Fano varieties (2009, background): “hat the pair (A2 F 3 ;2 3 div(xy) + 1 3 div(x2 y2)) is not sharply F-pure at the origin m = (x;y). To see this, using Fedder’s criterion for pairs, see [Sch08b, Theorem 4.1] (compare with [Fed83] and [Tak04a]), it is enough to observe that (xy)d(2)(3 e 1) 3 e(x2 y2)d(3 e 1) 3 e= (xy)(2)3e 1(x2 y2)3(”
A simple characterization of Du Bois singularities (2007, methodology): “It is also reasonable to ask whether the consequence of the adjunction-type theorem of 5.5 can be strengthened; see [Vas98] and [ Tak04 ].”
F-injective singularities are Du Bois (2008, background): “Further generalizations were made by Takagi; see [ Tak04 ] and [Tak08].”
ON THE BEHAVIOR OF TEST IDEALS UNDER FINITE MORPHISMS (2010, background): “.15. In case X is not the spectrum of a local ring (even in the case that X is aﬃne) the deﬁnitions of strongly F-regular and sharply F-pure triples given here diﬀer from those in the literature; see [Tak04a] and [Sch08]. However, the above deﬁnitions possess more desirable behavior with respect to ”
Integral closure of ideals, rings, and modules (2006, background): “precise in his paper [288] dealing with singularities of pairs (see also [290]).”
Test ideals in non-Q-Gorenstein rings (2009, unknown): “Deﬁnition 3.12 ([HH94], [HW02], [Tak04a]) .”