The valuative tree is the projective limit of Eggers-Wall trees

  1. González Pérez, Pedro D.
  2. García Barroso, Evelia R.
  3. Popescu-Pampu, Patrick
  1. 1 Universidad de La Laguna
    info

    Universidad de La Laguna

    San Cristobal de La Laguna, España

    ROR https://ror.org/01r9z8p25

  2. 2 Universidad Complutense de Madrid
    info

    Universidad Complutense de Madrid

    Madrid, España

    ROR 02p0gd045

Revista:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

ISSN: 1578-7303,  1579-1505

Año de publicación: 2019

Volumen: 113

Número: 4

Páginas: 4051-4105

Tipo: Artículo

DOI: 10.1007/S13398-019-00646-Z GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

Resumen

Consider a germ C of reduced curve on a smooth germ S of complex analytic surface. Assume that C contains a smooth branch L. Using the Newton-Puiseux series of C relative to any coordinate system (x, y) on S such that L is the y-axis, one may define the Eggers-Wall treeΘL(C) of C relative to L. Its ends are labeled by the branches of C and it is endowed with three natural functions measuring the characteristic exponents of the previous Newton-Puiseux series, their denominators and contact orders. The main objective of this paper is to embed canonically ΘL(C) into Favre and Jonsson’s valuative tree P(V) of real-valued semivaluations of S up to scalar multiplication, and to show that this embedding identifies the three natural functions on ΘL(C) as pullbacks of other naturally defined functions on P(V). As a consequence, we generalize the well-known inversion theorem for one branch: if L′ is a second smooth branch of C, then the valuative embeddings of the Eggers-Wall trees ΘL′(C) and ΘL(C) identify them canonically, their associated triples of functions being easily expressible in terms of each other. We prove also that the space P(V) is the projective limit of Eggers-Wall trees over all choices of curves C. As a supplementary result, we explain how to pass from ΘL(C) to an associated splice diagram.

Referencias bibliográficas

  • Abhyankar, S.S.: Inversion and invariance of characteristic pairs. Am. J. Maths. 89, 363–372 (1967)
  • Baker, M., Rumely, R.: Potential Theory and Dynamics on the Berkovich Projective Line. American Mathematical Society, Providence (2010)
  • Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 4. Compact Complex Surfaces. Springer, Berlin (2004)
  • Berkovich, V.G.: Spectral Theory and Analytic Geometry over Non-Archimedean Fields. American Mathematical Society, Providence (1990)
  • Boucksom, S., Favre, C., Jonsson, M.: Valuations and plurisubharmonic singularities. Publ. RIMS Kyoto Univ. 44, 449–494 (2008)
  • Corral, N.: Sur la topologie des courbes polaires de certains feuilletages singuliers. Ann. Inst. Fourier (Grenoble) 53(3), 787–814 (2003)
  • Defrise, P.: Étude locale des correspondances rationnelles entre surfaces algébriques. Mém. Soc. R. Sci. Liège 9(3), 133 (1949)
  • Deligne, P.: Intersections sur les surfaces régulières. In: Groupes de monodromie en géométrie algébrique. SGA 7 II, Lect. Notes in Maths. 340, Springer, Berlin, (1973), pp 1–37
  • Eggers, H.: Polarinvarianten und die Topologie von Kurvensingularitaeten. Bonner Math. Schriften 14, 147 (1983)
  • Eisenbud, D., Neumann, W.: Three-dimensional link theory and invariants of plane curve singularities. Princeton Univ. Press, Princeton (1985)
  • Favre, C., Jonsson, M.: The valuative tree. Lect. Notes in Maths. 1853. Springer, Berlin (2004)
  • Favre, C.: Holomorphic self-maps of singular rational surfaces. Publ. Mat. 54, 389–432 (2010)
  • García Barroso, E.R.: Invariants des singularités de courbes planes et courbure des fibres de Milnor. PhD thesis, Univ. La Laguna, Tenerife (Spain) (1996). http://ergarcia.webs.ull.es/tesis.pdf
  • García Barroso, E.R.: Sur les courbes polaires d’une courbe plane réduite. Proc. Lond. Math. Soc. 81, 1–28 (2000)
  • García Barroso, E.R., González Pérez, P.D.: Decomposition in bunches of the critical locus of a quasi-ordinary map. Compos. Math. 141, 461–486 (2005)
  • García Barroso, E.R., González Pérez, P.D., Popescu-Pampu, P.: Variations on inversion theorems for Newton-Puiseux series. Math. Ann. 368(3–4), 1359–1397 (2017)
  • García Barroso, E.R., González Pérez, P.D., Popescu-Pampu, P.: Ultrametric Spaces of Branches on Arborescent Singularities. In: Greuel, G.-M., Narváez, L., Xambó-Descamps, S. (eds.) Singularities, Algebraic Geometry, Commutative Algebra and Related Topics Festschrift for Antonio Campillo on the Occasion of his 65th Birthday, pp. 55–106. Springer, Berlin (2018)
  • García Barroso, E.R., González Pérez, P.D., Popescu-Pampu, P., Ruggiero, M.: Ultrametric properties for valuation spaces of normal surface singularities. Preprint (2018). arXiv:1802.01165
  • Gignac, W., Ruggiero, M.: Local dynamics of non-invertible maps near normal surface singularities, to appear in Mem. Amer. Math. Soc. Preprint (2018). arXiv:1704.04726v2
  • Guibert, G., Loeser, F., Merle, M.: Composition with a two variable function. Math. Res. Lett. 16(3), 439–448 (2009)
  • Halphen, G.: Sur une série de courbes analogues aux développées. J. Maths Pures Appliquées 2, 87–144 (1876)
  • Hocking, J.G., Young, G.S.: Topology. Constable and Company, Ltd., Edinburg (1961). (reprinted by Dover Publications 1988)
  • Jonsson, M.: Dynamics on Berkovich spaces in low dimensions. In Berkovich spaces and applications, 205–366. A. Ducros, C. Favre, J. Nicaise eds., Lect. Notes in Maths. 2119 (2015)
  • Kapranov, M.M.: Veronese curves and Grothendieck-Knudsen moduli space $$\overline{M}_{0, n}$$ M ¯ 0 , n . J. Algebraic Geom. 2(2), 239–262 (1993)
  • Kapranov, M.M.: The permutoassociahedron, Mac Lane’s coherence theorem and asymptotic zones for the KZ equation. J. Pure Appl. Algebra 85(2), 119–142 (1993)
  • Kuo, T.C., Lu, Y.C.: On analytic function germs of two complex variables. Topology 16(4), 299–310 (1977)
  • McNeal, J.D., Némethi, A.: The order of contact of a holomorphic ideal in $${\mathbb{C}}^2$$ C 2 . Math. Zeitschrift 250(4), 873–883 (2005)
  • Noether, M.: Les combinaisons caractéristiques dans la transformation d’un point singulier. Rend. Circ. Mat. Palermo IV 89–108, 300–301 (1890)
  • Novacoski, J.: Valuations centered at a two-dimensional regular local ring: infima and topologies., Valuation theory in interaction, EMS Ser. Congr. Rep., 389–403 (2014)
  • Orlik, P.: Seifert manifolds. Lecture Notes in Mathematics, vol. 291. Springer, Berlin-New York (1972)
  • Płoski, A.: Remarque sur la multiplicité d’intersection des branches planes. Bull. Polish Acad. Sci. Math. 33, 601–605 (1985)
  • Popescu-Pampu, P.: Arbres de contact des singularités quasi-ordinaires et graphes d’adjacence pour les 3-variétés réelles. Thèse, Univ. Paris 7, (2001). https://tel.archives-ouvertes.fr/tel-00002800v1
  • Popescu-Pampu, P.: Sur le contact d’une hypersurface quasi-ordinaire avec ses hypersurfaces polaires. J. Inst. Math. Jussieu 3, 105–138 (2004)
  • Seifert, H.: Topologie dreidimensionaler gefaserte Räume. Acta Math. 60, 147–238 (1932)
  • Seifert, H., Threlfall, W.: A Textbook of Topology. Academic Press, Cambridge (1980)
  • Shafarevich, I.: Basic Algebraic Geometry, vol. 1. Springer, Berlin (1994)
  • Siebenmann, L.: On vanishing of the Rohlin invariant and non-finitely amphicheiral homology $$3$$ 3 -spheres. In: Topology Symposium, Siegen, 1979, Koschorke and Neumann (eds.) Lecture Notes in Maths 788:172–222 (1979)
  • Smith, H.J.S.: On the higher singularities of plane curves. Proc. Lond. Math. Soc. VI, 153–182 (1875)
  • Stolz, O.: Die Multiplicität der Schnittpunkte zweier algebraischer Curven. Math. Ann. 15, 122–160 (1879)
  • Wall, C.T.C.: Chains on the Eggers tree and polar curves. In: Proc. of the Int. Conf. on Algebraic Geometry and Singularities (Sevilla, 2001). Rev. Mat. Iberoamericana 19(2):745–754 (2003)
  • Wall, C.T.C.: Singular points of plane curves. London Math. Society Student Texts 63. Cambridge Univ. Press, Cambridge (2004)
  • Zariski, O.: Studies in equisingularity III. Saturation of local rings and equisingularity. Am. J. Maths. 90, 961–1023 (1968)