Singularities of pairs via jet schemes
Author:
Mircea Mustata
Journal:
J. Amer. Math. Soc. 15 (2002), 599615
MSC (2000):
Primary 14B05; Secondary 14B10, 14E30
Published electronically:
February 14, 2002
MathSciNet review:
1896234
Fulltext 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. GuseĭnZade, 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
(86f:58018)
 [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
(2001a:14039)
 [Cr]
A. Craw, An introduction to motivic integration, preprint 1999, arXiv: math.AG/9911179.
 [DK]
J.P. Demailly and J. Kollár, Semicontinuity of complex singularity exponents and KahlerEinstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) 34 (2001), 525556.
 [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
(99k:14002), http://dx.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
(98m:14006)
 [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
(33 #7333)
 [Ho]
J.
A. Howald, Multiplier ideals of monomial
ideals, Trans. Amer. Math. Soc.
353 (2001), no. 7,
2665–2671 (electronic). MR 1828466
(2002b:14061), http://dx.doi.org/10.1090/S0002994701027209
 [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
(99m:14033)
 [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), 397424.
 [AGV]
 V. I. Arnol'd, S. M. GuseinZade and A. N. Varchenko, Singularities of differentiable maps, vol. I, Birkhäuser Boston Inc., Boston, MA, 1985. MR 86f:58018
 [Ba]
 V. V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, in Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 132, World Sci. Publishing, River Edge, NJ, 1998. MR 2001a:14039
 [Cr]
 A. Craw, An introduction to motivic integration, preprint 1999, arXiv: math.AG/9911179.
 [DK]
 J.P. Demailly and J. Kollár, Semicontinuity of complex singularity exponents and KahlerEinstein metrics on Fano orbifolds, Ann. Sci. École Norm. Sup. (4) 34 (2001), 525556.
 [DL]
 J. Denef and F. Loeser, Germs of arcs on singular varieties and motivic integration, Invent. Math. 135 (1999), 201232. MR 99k:14002
 [Ein]
 L. Ein, Multiplier ideals, vanishing theorems and applications, in Algebraic geometry, Santa Cruz 1995, 203220, volume 62 of Proc. Symp. Pure Math. Amer. Math. Soc. 1997. MR 98m:14006
 [Hi]
 H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. (2) 79 (1964), 109326. MR 33:7333
 [Ho]
 J. Howald, Multiplier ideals of monomial ideals, Trans. Amer. Math. Soc. 353 (2001), 26652671. MR 2002b:14061
 [Kol]
 J. Kollár, Singularities of pairs, in Algebraic geometry, Santa Cruz 1995, 221286, volume 62 of Proc. Symp. Pure Math. Amer. Math. Soc. 1997. MR 99m:14033
 [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), 397424.
Similar Articles
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:
http://dx.doi.org/10.1090/S0894034702003910
PII:
S 08940347(02)003910
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
