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• W2093747572 abstract "We investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to “F-singularities of pairs. The notions of F-regular and F-pure rings in characteristic <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are characterized by a splitting of the Frobenius map, and define some classes of rings having “mild singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolution of singularities in characteristic zero. These are defined also for pairs of a normal variety and a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Q> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-divisor on it, and play important roles in birational algebraic geometry. As an analog of these singularities of pairs, we introduce the concept of “F-singularities of pairs, namely strong F-regularity, divisorial F-regularity and F-purity for a pair <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper A comma normal upper Delta right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(A,Delta )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a normal ring <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of characteristic <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and an effective <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Q> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-divisor <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Delta> <mml:semantics> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:annotation encoding=application/x-tex>Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S p e c upper A> <mml:semantics> <mml:mrow> <mml:mi>Spec</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {Spec} A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The main theorem of this paper asserts that, if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K Subscript upper A Baseline plus normal upper Delta> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>K_{A}+Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Q> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Cartier, then the above three variants of F-singularities of pairs imply KLT, PLT and LC properties, respectively. We also prove some results for F-singularities of pairs which are analogous to singularities of pairs in characteristic zero." @default.
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• W2093747572 title "F-regular and F-pure rings vs. log terminal and log canonical singularities" @default.
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