Ultrametric properties for valuation spaces of normal surface singularities
| dc.contributor.author | García Barroso, Evelia R. | |
| dc.contributor.author | González Pérez, Pedro Daniel | |
| dc.contributor.author | Popescu-Pampu, Patrick | |
| dc.contributor.author | Ruggiero, Matteo | |
| dc.date.accessioned | 2023-06-17T12:45:55Z | |
| dc.date.available | 2023-06-17T12:45:55Z | |
| dc.date.issued | 2019 | |
| dc.description.abstract | Let L be a fixed branch – that is, an irreducible germ of curve – on a normal surface singularity X. If A, B are two other branches, define uL(A, B) := (L · A) (L · B) / A · B , where A · B denotes the intersection number of A and B. Call X arborescent if all the dual graphs of its good resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of Ploski by proving that whenever X is arborescent, the function uL is an ultrametric on the set of branches on X different from L. In the present paper we prove that, conversely, if uL is an ultrametric, then X is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on X, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which uL is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing L to be an arbitrary semivaluation on X and by defining uL on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if X is arborescent, and without any restriction on X we exhibit special subspaces of the space of semivaluations in restriction to which uL is still an ultrametric. | |
| dc.description.department | Depto. de Álgebra, Geometría y Topología | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | Ministerio de Economía, Industria y Competitividad (MINECO) | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/77900 | |
| dc.identifier.doi | 10.1090/tran/7854 | |
| dc.identifier.issn | 0002-9947 | |
| dc.identifier.officialurl | https://doi.org/10.1090/tran/7854 | |
| dc.identifier.relatedurl | https://arxiv.org/abs/1802.01165 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/12914 | |
| dc.issue.number | 12 | |
| dc.journal.title | Trans. Amer. Math. Soc. | |
| dc.language.iso | eng | |
| dc.page.final | 8475 | |
| dc.page.initial | 8423 | |
| dc.relation.projectID | MTM2016-80659-P; MTM2016-76868-C2-1-P | |
| dc.rights.accessRights | restricted access | |
| dc.subject.cdu | 512.7 | |
| dc.subject.keyword | Arborescent singularity | |
| dc.subject.keyword | B-divisor | |
| dc.subject.keyword | Birational geometry | |
| dc.subject.keyword | Block | |
| dc.subject.keyword | Brick | |
| dc.subject.keyword | Cut-vertex | |
| dc.subject.keyword | Cyclic element | |
| dc.subject.keyword | Intersection number | |
| dc.subject.keyword | Normal surface singularity | |
| dc.subject.keyword | Semivaluation | |
| dc.subject.keyword | Tree | |
| dc.subject.keyword | Ultrametric | |
| dc.subject.keyword | Valuation | |
| dc.subject.ucm | Geometria algebraica | |
| dc.subject.unesco | 1201.01 Geometría Algebraica | |
| dc.title | Ultrametric properties for valuation spaces of normal surface singularities | |
| dc.type | journal article | |
| dc.volume.number | 372 | |
| dcterms.references | [1] W. L. Ayres, Concerning continuous curves in metric space, Amer. J. Math. 51 (1929), no. 4, 577–594, DOI 10.2307/2370583. MR1507913 [2] Hans-Jürgen Bandelt and Michael Anthony Steel, Symmetric matrices representable by weighted trees over a cancellative abelian monoid, SIAM J. Discrete Math. 8 (1995), no. 4, 517–525, DOI 10.1137/S0895480191201759. MR1361386 [3] Sebastian Böcker and Andreas W. M. Dress, Recovering symbolically dated, rooted trees from symbolic ultrametrics, Adv. Math. 138 (1998), no. 1, 105–125, DOI 10.1006/aima.1998.1743. MR1645064 [4] J. A. Bondy and U. S. R. Murty, Graph theory, Graduate Texts in Mathematics, vol. 244, Springer, New York, 2008. MR2368647 [5] Sébastien Boucksom, Charles Favre, and Mattias Jonsson, Degree growth of meromorphic surface maps, Duke Math. J. 141 (2008), no. 3, 519–538, DOI 10.1215/00127094-2007-004. MR2387430 [6] Sébastien Boucksom, Charles Favre, and Mattias Jonsson, A refinement of Izumi’s theorem, Valuation theory in interaction, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2014, pp. 55–81. MR3329027 [7] Peter Buneman, A note on the metric properties of trees, J. Combinatorial Theory Ser. B 17 (1974), 48–50, DOI 10.1016/0095-8956(74)90047-1. MR0363963 [8] Ana Belén de Felipe, Topology of spaces of valuations and geometry of singularities, Trans. Amer. Math. Soc. 371 (2019), no. 5, 3593–3626, DOI 10.1090/tran/7441. MR3896123 [9] Patrick Du Val, On absolute and non absolute singularities of algebraic surfaces (English, with Turkish summary), Rev. Fac. Sci. Univ. Istanbul (A) 11 (1944), 159–215 (1946). MR0031286 [10] Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith, Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125 (2003), no. 2, 409–440. MR1963690 [11] Lorenzo Fantini, Normalized non-Archimedean links and surface singularities (English, with English and French summaries), C. R. Math. Acad. Sci. Paris 352 (2014), no. 9, 719–723, DOI 10.1016/j.crma.2014.06.010. MR3258263 [12] Lorenzo Fantini, Normalized Berkovich spaces and surface singularities, Trans. Amer. Math. Soc. 370 (2018), no. 11, 7815–7859, DOI 10.1090/tran/7209. MR3852450 [13] Charles Favre, Holomorphic self-maps of singular rational surfaces, Publ. Mat. 54 (2010), no. 2, 389–432, DOI 10.5565/PUBLMAT 54210 06. MR2675930 [14] Charles Favre and Mattias Jonsson, The valuative tree, Lecture Notes in Mathematics, vol. 1853, Springer-Verlag, Berlin, 2004. MR2097722 [15] Charles Favre and Mattias Jonsson, Eigenvaluations (English, with English and French summaries), Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 2, 309–349, DOI 10.1016/j.ansens.2007.01.002. MR2339287 [16] Charles Favre and Mattias Jonsson, Dynamical compactifications of C2, Ann. of Math. (2) 173 (2011), no. 1, 211–248, DOI 10.4007/annals.2011.173.1.6. MR2753603 [17] Robert L. Foote, A unified Pythagorean theorem in Euclidean, spherical, and hyperbolic geometries, Math. Mag. 90 (2017), no. 1, 59–69, DOI 10.4169/math.mag.90.1.59. MR3608701 [18] T. Gallai, Elementare Relationen bezüglich der Glieder und trennenden Punkte von Graphen (German, with Russian summary), Magyar Tud. Akad. Mat. Kutató Int. Közl. 9 (1964), 235–236. MR0169231 [19] Evelia R. Garcïa Barroso, Pedro D. González Pérez, and Patrick Popescu-Pampu, Ultrametric spaces of branches on arborescent singularities, Singularities, algebraic geometry, commutative algebra, and related topics, Springer, Cham, 2018, pp. 55–106. MR3839791 [20] William Gignac and Matteo Ruggiero, Growth of attraction rates for iterates of a superattracting germ in dimension two, Indiana Univ. Math. J. 63 (2014), no. 4, 1195–1234, DOI 10.1512/iumj.2014.63.5286. MR3263927 [21] W. Gignac and M. Ruggiero, Local dynamics of non-invertible maps near normal surface singularities, 2018, Mem. Amer. Math. Soc. (to appear). Preprint available at http://arxiv.org/abs/1704.04726, 2017. [22] Frank Harary, An elementary theorem on graphs, Amer. Math. Monthly 66 (1959), 405–407, DOI 10.2307/2308754. MR0104229 [23] Frank Harary, Graph theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969. MR0256911 [24] Frank Harary and Geert Prins, The block-cutpoint-tree of a graph, Publ. Math. Debrecen 13 (1966), 103–107. MR0211918 [25] Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, vol. 52, Springer-Verlag, New York-Heidelberg, 1977. MR0463157 [26] Ehud Hrushovski, François Loeser, and Bjorn Poonen, Berkovich spaces embed in Euclidean spaces, Enseign. Math. 60 (2014), no. 3-4, 273–292, DOI 10.4171/LEM/60-3/4-4. MR3342647 [27] Shuzo Izumi, Linear complementary inequalities for orders of germs of analytic functions, Invent. Math. 65 (1981/82), no. 3, 459–471, DOI 10.1007/BF01396630. MR643564 [28] Shuzo Izumi, A measure of integrity for local analytic algebras, Publ. Res. Inst. Math. Sci. 21 (1985), no. 4, 719–735, DOI 10.2977/prims/1195178926. MR817161 [29] Mattias Jonsson, Dynamics of Berkovich spaces in low dimensions, Berkovich spaces and applications, Lecture Notes in Math., vol. 2119, Springer, Cham, 2015, pp. 205–366, DOI 10.1007/978-3-319-11029-5 6. MR3330767 [30] Veerabhadra R. Kulli, The block-point tree of a graph, Indian J. Pure Appl. Math. 7 (1976), no. 6, 620–624. MR0505715 [31] K. Kuratowski and G. T. Whyburn. Sur les éléments cycliques et leurs applications, Fund. Math., 16 (1930), no. 1, 305–331. [32] B. Lehman, Cyclic element theory in connected and locally connected Hausdorff spaces, Canad. J. Math. 28 (1976), no. 5, 1032–1050, DOI 10.4153/CJM-1976-101-7. MR0420575 [33] Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR0276239 [34] Paolo Maraner, A spherical Pythagorean theorem, Math. Intelligencer 32 (2010), no. 3, 46–50, DOI 10.1007/s00283-010-9152-9. MR2721310 [35] B. L. McAllister, Cyclic elements in topology, a history, Amer. Math. Monthly 73 (1966), 337–350, DOI 10.2307/2315392. MR0200894 [36] Byron L. McAllister, A survey of cyclic element theory and recent developments, General topology and modern analysis (Proc. Conf., Univ. California, Riverside, Calif., 1980), Academic Press, New York-London, 1981, pp. 255–263. MR619050 [37] Mohammad Moghaddam, On Izumi’s theorem on comparison of valuations, Kodai Math. J. 34 (2011), no. 1, 16–30, DOI 10.2996/kmj/1301576758. MR2786777 [38] David Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 5–22. MR0153682 [39] J. Nikiel, H. M. Tuncali, and E. D. Tymchatyn, Continuous images of arcs and inverse limit methods, Mem. Amer. Math. Soc. 104 (1993), no. 498, viii+80, DOI 10.1090/memo/0498. MR1148017 [40] Arkadiusz Ploski, Remarque sur la multiplicité d’intersection des branches planes (French, with English and Russian summaries), Bull. Polish Acad. Sci. Math. 33 (1985), no. 11-12, 601–605 (1986). MR849408 [41] P. Popescu-Pampu, Ultrametric spaces of branches on arborescent singularities, Math. Forsch. Oberwolfach Report, 46 (2016), 2655–2658. [42] V. V. Prasolov and V. M. Tikhomirov, Geometry, translated from the 1997 Russian original by O. V. Sipacheva, Translations of Mathematical Monographs, vol. 200, American Mathematical Society, Providence, RI, 2001. MR1833867 [43] T. Radó and P. Reichelderfer, Cyclic transitivity, Duke Math. J. 6 (1940), 474–485. MR0001918 [44] John G. Ratcliffe, Foundations of hyperbolic manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 149, Springer, New York, 2006. MR2249478 [45] David Rees, Izumi’s theorem, Commutative algebra (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 407–416, DOI 10.1007/978-1-4612-3660-3 22. MR1015531 [46] G. Rond and M. Spivakovsky, The analogue of Izumi’s theorem for Abhyankar valuations, J. Lond. Math. Soc. (2) 90 (2014), no. 3, 725–740, DOI 10.1112/jlms/jdu045. MR3291797 [47] V. V. Shokurov, Prelimiting flips, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Linein. Sist. Konechno Porozhdennye Algebry, 82–219; English transl., Proc. Steklov Inst. Math. 1(240) (2003), 75–213. MR1993750 [48] Amaury Thuillier, Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels (French, with English summary), Manuscripta Math. 123 (2007), no. 4, 381–451, DOI 10.1007/s00229-007-0094-2. MR2320738 [49] W. T. Tutte, Connectivity in graphs, Mathematical Expositions, No. 15, University of Toronto Press, Toronto, Ont.; Oxford University Press, London, 1966. MR0210617 [50] Glen Van Brummelen, Heavenly mathematics: The forgotten art of spherical trigonometry, Princeton University Press, Princeton, NJ, 2013. MR3012466 [51] Michel Vaquié, Valuations (French), Resolution of singularities (Obergurgl, 1997), Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 539–590. MR1748635 [52] Tadé Wazewski, Sur les courbes de Jordan ne renfermant aucune courbe simple fermée de Jordan (French), NUMDAM, [place of publication not identified], 1923. 122pp. MR3532906 [53] Hassler Whitney, Non-separable and planar graphs, Trans. Amer. Math. Soc. 34 (1932), no. 2, 339–362, DOI 10.2307/1989545. MR1501641 [54] G. T. Whyburn, Cyclicly connected continuous curves, Proc. Natl. Acad. Sci. U.S.A., 13 (1927), no. 2, 31–38. [55] G. T. Whyburn, What is a curve?, Amer. Math. Monthly 49 (1942), 493–497, DOI 10.2307/2302854. MR0007109 [56] Gordon T. Whyburn, Cut points in general topological spaces, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 380–387, DOI 10.1073/pnas.61.2.380. MR0242129 [57] Oscar Zariski, The reduction of the singularities of an algebraic surface, Ann. of Math. (2) 40 (1939), 639–689, DOI 10.2307/1968949. MR0000159 [58] Oscar Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560–615, DOI 10.2307/1970376. MR0141668 | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | b7087753-f54f-4fdc-ac95-83b1b7fae921 | |
| relation.isAuthorOfPublication.latestForDiscovery | b7087753-f54f-4fdc-ac95-83b1b7fae921 |
