Singularities of pairs via jet schemes

By M. Mustaţă. 2001

164 citations recorded by the source corpus.

Let X be a smooth variety and Y a closed subscheme of X. By comparing motivic integrals on X and on a log resolution of (X,Y), we prove the following formula for the log canonical threshold of (X,Y): c(X,Y)=dim X-sup_m{(dim Y_m}/(m+1)}, where Y_m is the mth jet scheme of Y. We show how this formula can be used to study the log canonical threshold. In particular, we give a proof of the Semicontinuity theorem of Demailly and Koll\'ar.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f0a2f357b63bd5bee0d8adb184e6dc0128a32ed7

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In Geometry and Topology

1. S. Ishii
2. L. Ein
3. Karen E. Smith
4. Karl Schwede
5. M. Mustaţă
6. Tommaso de Fernex
7. R. Lazarsfeld
8. Dror Varolin

In Mathematical Physics

1. S. Ishii
2. L. Ein
3. Sumio Watanabe
4. J. Kollár
5. Karen E. Smith
6. R. Lazarsfeld
7. Tommaso de Fernex
8. M. Mustaţă

Citations in the corpus

• The Nash problem on arc families of singularities (2002, background): “It is also interesting to compare this with the results of [20], according to which the jet spaces Xm are usually reducible.”
• Jumping coefficients of multiplier ideals (2003, background): “Given an effective divisor A on X , recall that the log-canonical threshold of A at x is the positive (rational) number lct(A; x) = sup { c ∈ Q>0 | the pair (X , c ·A) is log-canonical at x } .”
• Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory (2010, background): “The real log canonical threshold λ is a birational invariant [Atiyah 70, Hiroanaka 64, Kashiwara 76, Koll´or et al. 98, Mustata 02, Watanabe 09] that represents the algebraic geometrical relation between the set of parameters W and the set of the optimal parameters W 0 .”
• The arc space of a toric variety (2003, background): “In fact, papers [5], [10], [11] by Mustaţǎ, Ein and Yasuda show that geometric properties of the jet schemes determine certain properties of the singularities of the base space.”
• A simple characterization of Du Bois singularities (2007, unknown): “We now discuss singularities associated to pairs ( Y, X ) where X has arbitrary codimension in Y ; see [Mus02] and [Tak04].”