$F$-singularities: applications of characteristic $p$ methods to singularity theory

DOI10.1090/suga/427zbMath1401.13016arXiv1409.3473OpenAlexW2963826629MaRDI QIDQ4568296

Shunsuke Takagi, Kei- ichi Watanabe

Publication date: 18 June 2018

Published in: Sugaku Expositions (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1409.3473

zbMATH Keywords

positive characteristictight closureFrobeniustest idealsHilbert-Kunz multiplicity

Mathematics Subject Classification ID

Singularities in algebraic geometry (14B05) Characteristic (p) methods (Frobenius endomorphism) and reduction to characteristic (p); tight closure (13A35)

Related Items

Local Cohomology—An Invitation, On the canonical, fpqc, and finite topologies on affine schemes. The state of the art, 𝐹-thresholds and test ideals of Thom-Sebastiani type polynomials, Singularities of determinantal pure pairs, Tate algebras and Frobenius non-splitting of excellent regular rings, Algebraic geometry in mixed characteristic, Inseparable maps on \(W_n\)-valued local cohomology groups of nontaut rational double point singularities and the height of K3 surfaces, Singularities in mixed characteristic. The perfectoid approach, Singularities and perfectoid geometry, GENERAL HYPERPLANE SECTIONS OF THREEFOLDS IN POSITIVE CHARACTERISTIC, THE GAMMA CONSTRUCTION AND ASYMPTOTIC INVARIANTS OF LINE BUNDLES OVER ARBITRARY FIELDS, FROBENIUS ACTIONS ON LOCAL COHOMOLOGY MODULES AND DEFORMATION

Cites Work

• Hilbert-Kunz Multiplicity of Three-Dimensional Local Rings
• A generalization of tight closure and multiplier ideals
• An interpretation of multiplier ideals via tight closure
• Strong F-regularity in images of regular rings
• Strong and weak F -regularity are equivalent for graded rings
• F-singularities via alterations
• On the -purity of isolated log canonical singularities
• On the behavior of test ideals under finite morphisms
• The non-nef locus in positive characteristic
• Characterizations of Regular Local Rings of Characteristic p
• MODULES WITH FINITE $F$-REPRESENTATION TYPE
• A CharacteristicpAnalog of Multiplier Ideals and Applications^#
• A characterization of rational singularities
• Uniform bounds and symbolic powers on smooth varieties
• Multiplicity and tight closures of parameters
• Globally F-regular varieties: Applications to vanishing theorems for quotients of Fano varieties
• F-rationality of Rees algebras
• On weak subintegrality
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Surfaces of globally \(F\)-regular and \(F\)-split type
• Adjoint ideals and a correspondence between log canonicity and \(F\)-purity
• Semi-log canonical vs \(F\)-pure singularities
• Test ideals of non-principal ideals: computations, jumping numbers, alterations and division theorems
• A canonical linear system associated to adjoint divisors in characteristic \(p>0\)
• \(F\)-signature exists
• Classification of the linearly reductive finite subgroup schemes of \(\mathrm{SL}_2\)
• Toroidal embeddings. I
• The rationality of the Hilbert-Kunz multiplicity in graded dimension two
• Discreteness and rationality of \(F\)-thresholds
• Rationality of Hilbert-Kunz multiplicities: a likely counterexample
• The singular Riemann-Roch theorem and Hilbert-Kunz functions
• Another proof of the global \(F\)-regularity of Schubert varieties
• Asymptotic invariants of base loci
• \(D\)-modules over rings with finite F-representation type
• \(F\)-adjunction
• Globally \(F\)-regular and log Fano varieties
• Tight closure does not commute with localization
• Centers of \(F\)-purity
• Generalized test ideals, sharp \(F\)-purity, and sharp test elements
• Ordinary reduction of K3 surfaces
• Singularités rationnelles et quotients par les groupes réductifs. (Rational singularities and quotients by reductive groups)
• Relèvements modulo \(p^ 2\) et décomposition du complexe de de Rham. (Lifting modulo \(p^ 2\) and decomposition of the de Rham complex)
• Rationale quasihomogene Singularitäten
• Pseudo-rational local rings and a theorem of Briancon-Skoda about integral closures of ideals
• Hodge cycles, motives, and Shimura varieties
• \(F\)-regular and \(F\)-pure normal graded rings
• Normal F-pure surface singularities
• The purity of the Frobenius and local cohomology
• The structure of one-dimensional F-pure rings
• On graded rings. I
• Some surprising Hilbert-Kunz functions
• \(F\)-rationality of certain Rees algebras and counterexamples to ``Boutot's theorem for \(F\)-rational rings
• Classification of two-dimensional F-regular and F-pure singularities
• The Hilbert-Kunz function of rings of finite Cohen-Macaulay type
• Notes on the Hilbert-Kunz function.
• Minimal relative Hilbert-Kunz multiplicity
• The \(F\)-signature and strong \(F\)-regularity.
• Hilbert-Kunz multiplicity and an inequality between multiplicity and colength
• Comparison of symbolic and ordinary powers of ideals.
• Semistability and Hilbert-Kunz multiplicities for curves
• On the upper semi-continuity of the Hilbert-Kunz multiplicity
• The Hilbert-Kunz function
• Two theorems about maximal Cohen-Macaulay modules
• Hilbert-Kunz functions for normal rings
• Characterization of log del Pezzo pairs via anticanonical models
• \(F\)-purity of hypersurfaces
• Surfaces of globally \(F\)-regular type are of Fano type
• A characteristic \(p\) analogue of plt singularities and adjoint ideals
• Resolution of singularities of an algebraic variety over a field of characteristic zero. I
• Inequalities related to certain couples of local rings
• Geometric interpretation of tight closure and test ideals
• The multiplier ideal is a universal test ideal
• On the commutation of the test ideal with localization and completion
• Hilbert-Kunz Multiplicity of Two-Dimensional Local Rings
• On arithmetic Macaulayfication of Noetherian rings
• Unmixed local rings with minimal Hilbert-Kunz multiplicity are regular
• New estimates of Hilbert–Kunz multiplicities for local rings of fixed dimension
• Ordinary varieties and the comparison between multiplier ideals and test ideals II
• Characterization of varieties of Fano type via singularities of Cox rings
• -PURITY VERSUS LOG CANONICITY FOR POLYNOMIALS
• F-Pure Homomorphisms, StrongF-regularity, andF-injectivity
• Ordinary varieties and the comparison between multiplier ideals and test ideals
• F-rational rings have rational singularities
• Global $F$-regularity of Schubert varieties with applications to $\mathcal {D}$-modules
• Formulas for multiplier ideals on singular varieties
• Tight closure and strong F-regularity
• Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
• Existence of minimal models for varieties of log general type
• Log canonical singularities are Du Bois
• A Simple Characterization of Du Bois Singularities
• Test Ideals Vs. Multiplier Ideals
• F-injective singularities are Du Bois
• F-Purity and Rational Singularity
• On Noetherian Rings of Characteristic p
• Cyclic Purity Versus Purity in Excellent Noetherian Rings
• Subadjunction of log canonical divisors, II
• A characterization of rational singularities in terms of injectivity of Frobenius maps
• Simplicity of Rings of Differential Operators in Prime Characteristic
• Some results on test elements
• F-regular and F-pure rings vs. log terminal and log canonical singularities
• The weak ordinarity conjecture and $F$-singularities
• On a generalization of test ideals