Publication: Motivic Poincaré series, toric singularities and logarithmic Jacobian ideals
Loading...
Full text at PDC
Publication Date
2012
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Mathematical Society
Citations
Exportar
Abstract
The geometric motivic Poincare 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.
Description
UCM subjects
Unesco subjects
Keywords
Citation
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., Zurich, 2006.
Brion, M., Polytopes convexes entiers. Gaz. Math. No. 67 (1996), 21–42.
Cobo Pablos, H., Arcos y series motıvicas de singularidades, Tesis Doctoral, Universidad Complutense de Madrid (2009).
Cobo Pablos, H., Gonzalez Perez, P.D., Geometric motivic Poincare series of quasi-ordinary hypersurfaces, Math.Proc. Camb. Phil. Soc. 149 (2010), no. 1, 49–74.
Denef, J. and Loeser. F., Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 135, 1, (1999), 201–232.
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,Birkhauser, Basel, 2001
Denef, J. and Loeser. F., Definable sets, motives and padic integrals, J. Amer.Math. Soc. 14, 4 (2001) 429–469. [D-L4] Denef, J. and Loeser. F., Motivic integration, quotient singularities and the McKay correspondance, Compositio Math. 131 (2002), 267–290.
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).
Ein, L. and Mustat¸a, M., Jet Schemes and Singularities,Algebraic geometry-Seattle 2005, 505–546, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc.,Providence, RI, 2009.
Ewald, G., Combinatorial Convexity and Algebraic Geometry, Springer-Verlag,1996.
Fulton, W., Introduction to Toric Varieties, Annals of Math. Studies (131), Princenton University Press, 1993. Z] Gel'fand, I.M., Kapranov, M.M. and Zelevinsky,A.V.,Discriminants, Resultants and Multi-Dimensional Determinants, Birkh¨auser, Boston, 1994.
Greenberg, M.J., Rational points in Henselian discrete valuation rings. Inst. Hautes Etudes Sci. Publ. Math. No. 31, 1966, 59–64.
Gonzalez Springberg, G., Transform´e de Nash et eventail de dimension 2. C. R.Acad. Sci. Paris Ser. A-B 284 (1977), no. 1, A69–A71.
Ishii, S., The arc space of a toric variety, J. Algebra,Volume 278 (2004), 666-683.
Ishii, S., Arcs, valuations and the Nash map. J. Reine Angew. Math. 588 (2005),71–92.
Ishii, S, Jet schemes, arc spaces and the Nash problem. C. R. Math. Acad. Sci. Soc.R. Can. 29 (2007), no. 1, 1–21
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.
Looijenga, E., Motivic measures, Seminaire Bourbaki,Expose 874, Asterisque 276,(2002), 267–297.
Nicaise, J., Motivic generating series for toric surface. singularities Math. Proc.Camb. Phil. Soc. 138 (2005), 383-400. [N2] Nicaise, J., Arcs and resolution of singularities Manuscripta Math. 116 (2005), 297-322.
Oda, T. , Convex Bodies and Algebraic Geometry, Annals of Math. Studies (131),Springer-Verlag, 1988.
Rond, G., Series de Poincare motiviques d’un germe d’hypersurface irreductible quasi-ordinaire, Asterisque. 157 (2008), 371–396.
Stanley, R.P., Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997.
Teissier, B., On the Semple-Nash modification, preprint 2005.