Jet schemes, log discrepancies and Inversion of Adjunction

Abstract

We use the theory of motivic integration for singular spaces to give a characterization of minimal log discrepencies in terms of the codimension of certain subsets of spaces of arcs. This is done for arbitrary pairs $(X,Y)$, with $X$ normal and Q-Gorenstein. As a first application, we prove a precise version of Inversion of Adjunction, in the case when the ambient variety is smooth. Another application concerns the semicontinuity of minimal log discrepancies on smooth varieties.

17 pages, AMS-LaTeX; v.2: the statement of Corollary 2.4 is corrected, Theorems 2.5 and 2.6 from the previous version have been combined in one stronger statement; v.3: final version, to appear in Invent. Math

References in corpus (3)

• Contact loci in arc spaces
• Dimensions of jet schemes of log singularities
• Inversion of adjunction for non-degenerate hypersurfaces

Cited by in corpus (31)

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• Arcs on Determinantal Varieties
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• The Kodaira dimension of spaces of rational curves on low degree hypersurfaces
• On semi-continuity problems for minimal log discrepancies
• A boundedness conjecture for minimal log discrepancies on a fixed germ
• Wilder McKay correspondences
• Differentials on the arc space
• Inversion of adjunction for quotient singularities
• The minimal log discrepancy
• On the log minimal model program for irreducible symplectic varieties
• Isomorphisms of jet schemes
• Log canonical thresholds in positive characteristic
• On a comparison of minimal log discrepancies in terms of motivic integration
• Birationally rigid hypersurfaces
• Multiplicity of jet schemes of monomial schemes
• Nash multiplicity sequences and Hironaka's order function
• Nash multiplicities and resolution invariants
• On termination of flips and exceptionally non-canonical singularities
• Ideal-adic semi-continuity problem for minimal log discrepancies
• Ideal-adic semi-continuity of minimal log discrepancies on surfaces
• Minimal log discrepancies in positive characteristic
• Mather discrepancy and the arc spaces
• Stringy E-functions of hypersurfaces and of Brieskorn singularities
• Towards boundedness of minimal log discrepancies by Riemann--Roch theorem
• On minimal models of projective Hyperkaehler manifolds
• Jet schemes and singularities of W^r_d(C) loci
• On higher dimensional complex Plateau problem
• Smoothness and jet schemes
• A number theoretic characterization of $E$-smooth and (FRS) morphisms: estimates on the number of $\mathbb{Z}/p^k\mathbb{Z}$-points
• Smooth solution to higher dimensional complex Plateau problem