Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Motivic Poincaré series, toric singularities and logarithmic Jacobian ideals

Authors: H. Cobo Pablos and P. D. González Pérez
Journal: J. Algebraic Geom. 21 (2012), 495-529
Published electronically: August 22, 2011
MathSciNet review: 2914802
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Abstract | References | Additional Information

Abstract: The geometric motivic Poincaré series of a variety, which was introduced by Denef and Loeser, takes into account the classes in the Grothendieck ring of the sequence of jets of arcs in the variety. Denef and Loeser proved that this series has a rational form. We describe it in the case of an affine toric variety of arbitrary dimension. The result, which provides an explicit set of candidate poles, is expressed in terms of the sequence of Newton polyhedra of certain monomial ideals, which we call logarithmic Jacobian ideals, associated to the modules of differential forms with logarithmic poles outside the torus of the toric variety.

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  • [B-P] BARVINOK, A., The complexity of generating functions for integer points in polyhedra and beyond. International Congress of Mathematicians. Vol. III, 763-787, Eur. Math. Soc., Zürich, 2006. MR 2275706 (2007k:05007)
  • [Br] BRION, M., Polytopes convexes entiers. Gaz. Math. No. 67 (1996), 21-42. MR 1378294 (97d:52022)
  • [C] COBO PABLOS, H., Arcos y series motívicas de singularidades, Tesis Doctoral, Universidad Complutense de Madrid (2009).
  • [C-GP] COBO PABLOS, H., GONZÁLEZ EREZ, P.D., Geometric motivic Poincaré series of quasi-ordinary hypersurfaces, Math. Proc. Camb. Phil. Soc. 149 (2010), no. 1, 49-74. MR 2651577
  • [D-L1] DENEF, J. AND LOESER. F., Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 135, 1, (1999), 201-232. MR 1664700 (99k:14002)
  • [D-L2] DENEF, J. AND LOESER. F., Geometry on arc spaces of algebraic varieties. European Congress of Mathematics, Vol. I (Barcelona, 2000), 327-348, Progr. Math., 201, Birkhäuser, Basel, 2001 MR 1905328 (2004c:14037)
  • [D-L3] DENEF, J. AND LOESER. F., Definable sets, motives and $ p$-adic integrals, J. Amer. Math. Soc. 14, 4 (2001) 429-469. MR 1815218 (2002k:14033)
  • [D-L4] DENEF, J. AND LOESER. F., Motivic integration, quotient singularities and the McKay correspondance, Compositio Math. 131 (2002), 267-290. MR 1905024 (2004e:14010)
  • [D-L5] DENEF, J. AND LOESER. F., On some rational generating series occuring in arithmetic geometry, in Geometric Aspects of Dwork Theory, edited by A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz and F. Loeser, volume 1, de Gruyter, 509-526 (2004). MR 2099079 (2005h:11267)
  • [E-M] EIN, L. AND MUSTA¸TA, M., Jet Schemes and Singularities, Algebraic geometry-Seattle 2005, 505-546, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009. MR 2483946 (2010h:14004)
  • [Ew] EWALD, G., Combinatorial Convexity and Algebraic Geometry, Springer-Verlag, 1996. MR 1418400 (97i:52012)
  • [F] FULTON, W., Introduction to Toric Varieties, Annals of Math. Studies (131), Princenton University Press, 1993. MR 1234037 (94g:14028)
  • [GKZ] GEL'FAND, I.M., KAPRANOV, M.M. AND ZELEVINSKY, A.V., Discriminants, Resultants and Multi-Dimensional Determinants, Birkhäuser, Boston, 1994. MR 1264417 (95e:14045)
  • [Gr] GREENBERG, M.J., Rational points in Henselian discrete valuation rings. Inst. Hautes Études Sci. Publ. Math. No. 31, 1966, 59-64. MR 0207700 (34:7515)
  • [GS] GONZALEZ SPRINGBERG, G., Transformé de Nash et éventail de dimension $ 2$. C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 1, A69-A71. MR 0429894 (55:2903)
  • [I1] ISHII, S., The arc space of a toric variety, J. Algebra, Volume 278 (2004), 666-683. MR 2071659 (2005f:14100)
  • [I2] ISHII, S., Arcs, valuations and the Nash map. J. Reine Angew. Math. 588 (2005), 71-92. MR 2196729 (2006k:14005)
  • [I3] ISHII, S, Jet schemes, arc spaces and the Nash problem. C. R. Math. Acad. Sci. Soc. R. Can. 29 (2007), no. 1, 1-21 MR 2354631 (2008j:14019)
  • [LJ-R] LEJEUNE-JALABERT, M. AND REGUERA, A., The Denef-Loeser series for toric surface singularities. Proceedings of the International Conference on Algebraic Geometry and Singularities (Spanish) (Sevilla, 2001). Rev. Mat. Iberoamericana 19 (2003), no. 2, 581-612. MR 2023199 (2004m:14001)
  • [Lo] LOOIJENGA, E., Motivic measures, Séminaire Bourbaki, Exposé 874, Astérisque 276, (2002), 267-297. MR 1886763 (2003k:14010)
  • [N1] NICAISE, J., Motivic generating series for toric surface. singularities Math. Proc. Camb. Phil. Soc. 138 (2005), 383-400. MR 2138569 (2006d:14012)
  • [N2] NICAISE, J., Arcs and resolution of singularities Manuscripta Math. 116 (2005), 297-322. MR 2130945 (2006h:14019)
  • [O] ODA, T. , Convex Bodies and Algebraic Geometry, Annals of Math. Studies (131), Springer-Verlag, 1988. MR 922894 (88m:14038)
  • [R] ROND, G., Séries de Poincaré motiviques d'un germe d'hypersurface irréductible quasi-ordinaire, Astérisque. 157 (2008), 371-396. MR 2647979 (2011c:14039)
  • [S] STANLEY, R.P., Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. MR 1442260 (98a:05001)
  • [T] TEISSIER, B., On the Semple-Nash modification, preprint 2005.
  • [V] VEYS, W., Arc spaces, motivic integration and stringy invariants, Advanced Studies in Pure Mathematics 43, Proceedings of ``Singularity Theory and its applications, Sapporo (Japan), 16-25 September 2003'' (2006), 529-572. MR 2325153 (2008g:14023)

Additional Information

H. Cobo Pablos
Affiliation: Depto. Algebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de las Ciencias 3. 28040, Madrid, Spain

P. D. González Pérez
Affiliation: Instituto de Ciencias Matemáticas-CSIC-UAM-UC3M-UCM, Depto. Algebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de las Ciencias 3. 28040, Madrid, Spain

Received by editor(s): September 17, 2009
Received by editor(s) in revised form: March 18, 2010
Published electronically: August 22, 2011
Additional Notes: The first author is supported by a grant of Fundación Caja Madrid. The second author is supported by Programa Ramón y Cajal of Ministerio de Educación y Ciencia (MEC), Spain. Both authors are supported by MTM2007-6798-C02-02 grant of MEC

American Mathematical Society