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Idealistic exponents: tangent cone, ridge, characteristic polyhedra. (English) Zbl 07262038
Summary: We study Hironaka’s idealistic exponents over \(\text{Spec} (\mathbb{Z})\). We give an idealistic interpretation of the tangent cone, the directrix, and the ridge. The main purpose is to introduce the notion of characteristic polyhedra of idealistic exponents and deduce from them intrinsic data on the idealistic exponent.
MSC:
14B05 Singularities in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
32S Complex singularities
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[1] Aroca, J.; Hironaka, H.; Vincente, J., The Theory of the Maximal Contact, Memorias de Matemática del Instituto “Jorge Juan”, vol. 29 (1975), Instituto “Jorge Juan” de Matemáticas, Consejo Superior de Investigaciones Cientificas: Instituto “Jorge Juan” de Matemáticas, Consejo Superior de Investigaciones Cientificas Madrid
[2] Berthomieu, J.; Hivert, P.; Mourtada, H., Computing Hironaka’s invariants: ridge and directrix, (Arithmetic, Geometry, Cryptography and Coding Theory 2009. Arithmetic, Geometry, Cryptography and Coding Theory 2009, Contemp. Math., vol. 521 (2010), Amer. Math. Soc.), 9-20 · Zbl 1246.14022
[3] Bierstone, E.; Milman, P., Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math., 128, 207-302 (1997) · Zbl 0896.14006
[4] Bierstone, E.; Milman, P., Desingularization algorithms I, role of exceptional divisors, Mosc. Math. J., 3, 751-805 (2003) · Zbl 1052.14019
[5] Benito, A.; Villamayor, O., On elimination of variables in the study of singularities in positive characteristic, Indiana Univ. Math. J., 64, 357-410 (2015) · Zbl 1318.14013
[6] Bravo, A.; García Escamilla, M. L.; Villamayor, O., On Rees algebras and invariants of singularities over perfect fields, Indiana Univ. Math. J., 61, 1201-1251 (2012) · Zbl 1315.14023
[7] Cossart, V., Sur le polyèdre caractéristique d’une singularité, Bull. Soc. Math. Fr., 103, 13-19 (1975) · Zbl 0333.32008
[8] Cossart, V., Polyèdre caractéristique et éclatements combinatoires, Rev. Mat. Iberoam., 5, 67-95 (1989) · Zbl 0708.14010
[9] Cossart, V., Is there a notion of weak maximal contact in characteristic \(p > 0\)?, Asian J. Math., 15, 3, 357-369 (2011) · Zbl 1264.14022
[10] Cossart, V.; Giraud, J.; Orbanz, U., Resolution of Surface Singularities, Lecture Notes in Mathematics, vol. 1101 (1984), Springer Verlag, with an appendix by H. Hironaka
[11] Cossart, V.; Jannsen, U.; Saito, S., Desingularization: Invariants and Strategy. Application to Dimension 2, Lecture Notes in Mathematics, vol. 2270 (2020), Springer Verlag, with an appendix by B. Schober · Zbl 07248448
[12] Cossart, V.; Jannsen, U.; Schober, B., Invariance of Hironaka’s characteristic polyhedron, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 113, 4, 4145-4169 (2019) · Zbl 1430.14077
[13] Cossart, V.; Piltant, O.; Schober, B., Faîte du cône tangent à une singularité: un théorème oublié, C. R. Acad. Sci. Paris, Ser. I, 355, 455-459 (2017) · Zbl 1360.14011
[14] Cossart, V.; Piltant, O., Characteristic polyhedra of singularities without completion, Math. Ann., 361, 157-167 (2015) · Zbl 1308.14008
[15] Cossart, V.; Piltant, O., Resolution of singularities of threefolds in positive characteristic II, J. Algebra, 321, 1836-1976 (2009) · Zbl 1173.14012
[16] Cossart, V.; Piltant, O., Resolution of singularities of arithmetical threefolds, J. Algebra, 529, 268-535 (2019) · Zbl 07057272
[17] Cossart, V.; Schober, B., Characteristic polyhedra of singularities without completion: part II, Collect. Math. (2020)
[18] Cossart, V.; Schober, B., A strictly decreasing invariant for resolution of singularities in dimension two, Publ. Res. Inst. Math. Sci., 56, 4, 4145-4169 (2020) · Zbl 1430.14077
[19] Giraud, J., Étude locale des singularités, Cours de \(3^{\text{ème}}\) cycle, 1971-1972, Publ. Math. (d’Orsay), 26 (1972) · Zbl 0394.14005
[20] Giraud, J., Sur la théorie du contact maximal, Math. Z., 137, 285-310 (1974) · Zbl 0275.32003
[21] Giraud, J., Contact maximal en caractéristique positive, Ann. Sci. Éc. Norm. Supér. (4), 8, 2, 201-234 (1975) · Zbl 0306.14004
[22] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero I + II, Ann. Math., 79, 109-326 (1964) · Zbl 0122.38603
[23] Hironaka, H., Characteristic polyhedra of singularities, J. Math. Kyoto Univ., 7, 251-293 (1967) · Zbl 0159.50502
[24] Hironaka, H., Idealistic exponents of singularity, (Algebraic Geometry (1977), Johns Hopkins Univ. Press), 52-125 · Zbl 0496.14011
[25] Hironaka, H., Desingularization of excellent surfaces, (Adv. Sci. Sem. in Alg. Geo. Adv. Sci. Sem. in Alg. Geo, Bowdoin College, Summer (1967)), Notes by Bruce Bennett, appendix of [10] (1984)
[26] Hironaka, H., Theory of infinitely near singular points, J. Korean Math. Soc., 40, 901-920 (2003) · Zbl 1055.14013
[27] Hironaka, H., Three key theorems on infinitely near singularities, (Singularités Franco-Japonaises. Singularités Franco-Japonaises, Sémin. Congr., vol. 10 (2005), Soc. Math. France: Soc. Math. France Paris), 87-126 · Zbl 1093.14021
[28] Kawanoue, H., Introduction to the idealistic filtration program with emphasis on the radical saturation, (The Resolution of Singular Algebraic Varieties (2014), Amer. Math. Soc.), 285-317 · Zbl 1326.14030
[29] Mourtada, H.; Schober, B., A polyhedral characterization of quasi-ordinary singularities, Mosc. Math. J., 18, 4, 755-785 (2018) · Zbl 1420.14008
[30] Narasimhan, R., Hyperplanarity of the equimultiple locus, Proc. Am. Math. Soc., 87, 3, 403-408 (1983) · Zbl 0521.13014
[31] Schober, B., Characteristic polyhedra of idealistic exponents with history (2013), Universität Regensburg, Dissertation
[32] Schober, B., A polyhedral approach to the invariant of Bierstone and Milman (2014), preprint, available on
[33] Schober, B., Partial local resolution by characteristic zero methods, Results Math., 73, 48, 39 pp (2018) · Zbl 1391.14071
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