Singularities of pairs via jet schemes

Author:
Mircea Mustata

Journal:
J. Amer. Math. Soc. **15** (2002), 599-615

MSC (2000):
Primary 14B05; Secondary 14B10, 14E30

DOI:
https://doi.org/10.1090/S0894-0347-02-00391-0

Published electronically:
February 14, 2002

MathSciNet review:
1896234

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a smooth variety and a closed subscheme. We use motivic integration on the space of arcs of to characterize the fact that is log canonical or log terminal using the dimension of the jet schemes of . This gives a formula for the log canonical threshold of , which we use to prove a result of Demailly and Kollár on the semicontinuity of log canonical thresholds.

**[AGV]**V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko,*Singularities of differentiable maps. Vol. I*, Monographs in Mathematics, vol. 82, Birkhäuser Boston, Inc., Boston, MA, 1985. The classification of critical points, caustics and wave fronts; Translated from the Russian by Ian Porteous and Mark Reynolds. MR**777682****[Ba]**Victor V. Batyrev,*Stringy Hodge numbers of varieties with Gorenstein canonical singularities*, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 1–32. MR**1672108****[Cr]**A. Craw,*An introduction to motivic integration*, preprint 1999, arXiv: math.AG/9911179.**[DK]**J.-P. Demailly and J. Kollár,*Semi-continuity of complex singularity exponents and Kahler-Einstein metrics on Fano orbifolds*, Ann. Sci. École Norm. Sup. (4)**34**(2001), 525-556.**[DL]**Jan Denef and François Loeser,*Germs of arcs on singular algebraic varieties and motivic integration*, Invent. Math.**135**(1999), no. 1, 201–232. MR**1664700**, https://doi.org/10.1007/s002220050284**[Ein]**Lawrence Ein,*Multiplier ideals, vanishing theorems and applications*, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 203–219. MR**1492524****[Hi]**Heisuke Hironaka,*Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II*, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2)**79**(1964), 205–326. MR**0199184****[Ho]**J. A. Howald,*Multiplier ideals of monomial ideals*, Trans. Amer. Math. Soc.**353**(2001), no. 7, 2665–2671. MR**1828466**, https://doi.org/10.1090/S0002-9947-01-02720-9**[Kol]**János Kollár,*Singularities of pairs*, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221–287. MR**1492525****[Kon]**M. Kontsevich, Lecture at Orsay (December 7, 1995).**[La]**R. Lazarsfeld,*Multiplier ideals for algebraic geometers*, lecture notes available at http://www.math.lsa.umich.edu/~rlaz, version of August 2000.**[Mu]**M. Mustata,*Jet schemes of locally complete intersection canonical singularities*, with an appendix by David Eisenbud and Edward Frenkel, Invent. Math.**145**(2001), 397-424.

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2000):
14B05,
14B10,
14E30

Retrieve articles in all journals with MSC (2000): 14B05, 14B10, 14E30

Additional Information

**Mircea Mustata**

Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720 – and – Institute of Mathematics of the Romanian Academy

Address at time of publication:
Clay Mathematics Institute, 1770 Massachusetts Avenue, No. 331, Cambridge, Massachusetts 02140

Email:
mirceamustata@yahoo.com

DOI:
https://doi.org/10.1090/S0894-0347-02-00391-0

Keywords:
Jet schemes,
log canonical threshold,
motivic integration

Received by editor(s):
March 2, 2001

Published electronically:
February 14, 2002

Article copyright:
© Copyright 2002
American Mathematical Society