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

2003/05/31 by Shunsuke Takagi · 2 citations
Mathematics · #Adjunction #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Codimension #Combinatorics #Commutative Algebra and Its Applications #Exponent #Geometry #Gravitational singularity #Inversion (geology) #Mathematical analysis #Mathematics #Pure mathematics #Subvariety #Variety (cybernetics) #math.AC #math.AG #msc:13A35 #msc:14B05 #msc:14E15 #msc:14E30

paper · pdf · doi:10.1007/s00222-003-0350-3

21 pages, AMS-LaTeX; v.2: minor changes, to appear in Invent. Math

arxiv created 2003/12/03 · openalex publication_date 2004/06/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We generalize the notions of F-regular and F-pure rings to pairs (R,\a^t) of rings R and ideals \a ⊂ 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 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.

Citations

• Positivity in Algebraic Geometry I
• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I

Cited by

• On the behavior of test ideals under finite morphisms
• Globally F-regular and log Fano varieties