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Local polar varieties in the geometric study of singularities
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 4, pp. 679-775.

Ce texte présente plusieurs aspects de la théorie de l’équisingularité des espaces analytiques complexes telle qu’elle est définie par les conditions de Whitney. Le but est de décrire des points de vue géométrique, topologique et algébrique une partition canonique localement finie d’un espace analytique complexe réduit X en strates non singulières telles que la géométrie locale de X soit constante le long de chaque strate. Les variétés polaires locales apparaissent dans le titre parce qu’elles jouent un rôle central dans l’unification des points de vue. Le point de vue géométrique conduit à l’étude des directions limites en un point donné de X n des hyperplans de n tangents à X en des points non singuliers. Ceci amène à réaliser que les conditions de Whitney, qui servent à définir la stratification, sont en fait de nature lagrangienne. Les variétés polaires locales sont utilisées pour analyser la structure de l’ensemble des positions limites d’hyperplans tangents. Cette structure aide à comprendre comment une singularité diffère de son cône tangent, supposé réduit. Les multiplicités des variétés polaires locales sont reliées à des invariants topologiques locaux, des caractéristiques d’Euler–Poincaré évanescentes, par une formule qui se révèle, dans le cas particulier où la singularité est le sommet du cône sur une variété projective réduite, donner une formule du type Plücker pour le calcul du degré de la variété duale d’une variété projective.

This text presents several aspects of the theory of equisingularity of complex analytic spaces from the standpoint of Whitney conditions. The goal is to describe from the geometrical, topological, and algebraic viewpoints a canonical locally finite partition of a reduced complex analytic space X into nonsingular strata with the property that the local geometry of X is constant on each stratum. Local polar varieties appear in the title because they play a central role in the unification of viewpoints. The geometrical viewpoint leads to the study of spaces of limit directions at a given point of X n of hyperplanes of n tangent to X at nonsingular points, which in turn leads to the realization that the Whitney conditions, which are used to define the stratification, are in fact of a Lagrangian nature. The local polar varieties are used to analyze the structure of the set of limit directions of tangent hyperplanes. This structure helps in particular to understand how a singularity differs from its tangent cone, assumed to be reduced. The multiplicities of local polar varieties are related to local topological invariants, local vanishing Euler–Poincaré characteristics, by a formula which turns out to contain, in the special case where the singularity is the vertex of the cone over a reduced projective variety, a Plücker-type formula for the degree of the dual of a projective variety.

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DOI : https://doi.org/10.5802/afst.1582
@article{AFST_2018_6_27_4_679_0,
     author = {Arturo Giles Flores and Bernard Teissier},
     title = {Local polar varieties in the geometric study of singularities},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {679--775},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {4},
     year = {2018},
     doi = {10.5802/afst.1582},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1582/}
}
Arturo Giles Flores; Bernard Teissier. Local polar varieties in the geometric study of singularities. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 4, pp. 679-775. doi : 10.5802/afst.1582. https://afst.centre-mersenne.org/articles/10.5802/afst.1582/

[1] Eric Dago Akéké Equisingular generic discriminants and Whitney conditions, Ann. Fac. Sci. Toulouse, Math., Volume 17 (2008) no. 4, pp. 661-671 | Zbl 1171.32015

[2] Clementa Alonso; Nuria Corral; Dũng Tràng Lê Limites de espacios tangentes en superficies, Monografías del Seminario Iberoamericano de Matemáticas, Volume 1, Instituto Interuniversitario de Estudios de Iberoamerica y Portugal, 2002, 128 pages | Zbl 1031.32020

[3] Paolo Aluffi On the singular schemes of hypersurfaces, Duke Math. J., Volume 80 (1995) no. 2, pp. 325-351 | Zbl 0876.14028

[4] Paolo Aluffi Euler characteristics of general linear sections and polynomial Chern classes, Rend. Circ. Mat. Palermo, Volume 62 (2013) no. 1, pp. 3-26 | Zbl 1312.14018

[5] Paolo Aluffi Projective duality and a Chern-Mather involution, Trans. Am. Math. Soc., Volume 370 (2018) no. 3, pp. 1803-1822 | Zbl 06823248

[6] Paolo Aluffi; Jean-Paul Brasselet Interpolation of characteristic classes of singular hypersurfaces, Adv. Math., Volume 180 (2003) no. 2, pp. 692-704 | Zbl 1073.14504

[7] Michael F. Atiyah; Ian G. Macdonald Introduction to Commutative Algebra, Addison-Wesley Publishing Company, 1969 | Zbl 0175.03601

[8] Bernd Bank; Marc Giusti; Joos Heintz; Guy Merlin Mbakop Polar varieties, real equation solving and data structures, J. Complexity, Volume 13 (1997) no. 1, pp. 5-27 | Zbl 0872.68066

[9] Bernd Bank; Marc Giusti; Joos Heintz; Guy Merlin Mbakop Polar varieties and efficient real elimination, Math. Z., Volume 238 (2001) no. 1, pp. 115-144 | Zbl 1073.14554

[10] Bernd Bank; Marc Giusti; Joos and Heintz Generalized polar varieties: geometry and algorithms, J. Complexity, Volume 21 (2005) no. 4, pp. 377-412 | Zbl 1085.14047

[11] Lev Birbrair; Alexandre Fernandes; Dũng Tràng Lê; J. Edson Sampaio Lipschitz regular complex algebraic sets are smooth, Proc. Am. Math. Soc., Volume 144 (2016) no. 3, pp. 983-987 | Zbl 1338.14008

[12] Romain Bondil Fine polar invariants of minimal surface singularities, J. Singul., Volume 14 (2016), pp. 91-112 | Zbl 1348.32001

[13] Nicolas Bourbaki Elements de Mathématique. Algèbre Commutative, Chap. I-VII, Masson, 1983

[14] Nicolas Bourbaki Elements de Mathématique. Algèbre Commutative, Chap. VIII-IX, Masson, 1983 | Zbl 0579.13001

[15] Jean-Paul Brasselet Milnor classes via polar varieties, Singularities in algebraic and analytic geometry (Contemporary Mathematics) Volume 266, American Mathematical Society, 2000, pp. 181-187 | Zbl 0991.14003

[16] Jean-Paul Brasselet The Schwartz classes of complex analytic singular varieties, Singularity theory, World Scientific, 2007, pp. 3-32 | Zbl 1124.14008

[17] Joël Briançon; Jean-Pierre-Georges Henry Equisingularité générique des familles de surfaces à singularité isolée, Bull. Soc. Math. Fr., Volume 108 (1980), pp. 259-281 | Zbl 0482.14004

[18] Joël Briancon; Jean-Paul Speder La trivialité topologique n’implique pas les conditions de Whitney, C. R. Acad. Sci., Paris, Sér. A, Volume 280 (1975) no. 6, p. A365-A367 | Zbl 0331.32010

[19] Joël Briancon; Jean-Paul Speder Les conditions de Whiney impliquent μ * constant, Ann. Inst. Fourier, Volume 26 (1976) no. 2, pp. 153-163 | Zbl 0331.32012

[20] Hans Brodersen; David Trotman Whitney (b) regularity is weaker than Kuo’s ratio test for real algebraic stratifications, Math. Scand., Volume 45 (1979), pp. 27-34 | Zbl 0429.58001

[21] Peter Bürgisser; Martin Lotz The complexity of computing the Hilbert polynomial of smooth equidimensional complex projective varieties, Found. Comput. Math., Volume 7 (2007) no. 1, pp. 51-86 | Zbl 1108.68059

[22] Fabrizio Catanese; Cecilia Trifogli Focal loci of algebraic varieties. I, Commun. Algebra, Volume 28 (2000) no. 12, pp. 6017-6057 | Zbl 1011.14001

[23] Denis Cheniot Sur les sections transversales d’un ensemble stratifié, C. R. Acad. Sci., Paris, Sér. A, Volume 275 (1972), p. 915-916 | Zbl 0249.32008

[24] Georges Comte Multiplicity of complex analytic sets and bi-Lipschitz maps, Real analytic and algebraic singularities (Pitman Research Notes in Mathematics Series) Volume 381, Longman, 1998, pp. 182-188 | Zbl 0982.32026

[25] Georges Comte Equisingularité réelle, nombres de Lelong, et images polaires, Ann. Sci. Éc. Norm. Supér., Volume 33 (2000) no. 6, pp. 757-788 | Zbl 0981.32018

[26] Georges Comte; Michel Merle Équisingularité réelle, II. Invariants locaux et conditions de régularité, Ann. Sci. Éc. Norm. Supér., Volume 41 (2008) no. 2, pp. 221-269 | Zbl 1163.32012

[27] Nuria Corral Sur la topologie des courbes polaires de certains feuilletages singuliers, Ann. Inst. Fourier, Volume 53 (2003) no. 3, pp. 787-814 | Zbl 1032.32019

[28] Jan Draisma; Emil Horobeţ; Giorgio Ottaviani; Bernd Sturmfels; Rekha R. Thomas The Euclidean Distance Degree of an Algebraic Variety, Found. Comput. Math., Volume 16 (2016) no. 1, pp. 99-149 | Zbl 1370.51020

[29] Lawrence Ein The ramification divisor for branched coverings of P n , Math. Ann., Volume 261 (1982), pp. 483-485 | Zbl 0519.14005

[30] David Eisenbud Commutative Algebra with a view toward Algebraic Geometry, Graduate Texts in Mathematics, Volume 150, Springer, 1999, xvi+785 pages | Zbl 0819.13001

[31] Lars Ernström A Plücker formula for singular projective varieties, Commun. Algebra, Volume 25 (1997) no. 9, pp. 2897-2901 | Article | Zbl 0891.14012

[32] Alexandre Fernandes; J. Edson Sampaio Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomorphisms, J. Topol., Volume 9 (2016) no. 3, pp. 927-933 | Zbl 1353.14005

[33] Gerd Fischer Complex Analytic Geometry, Lecture Notes in Mathematics, Volume 538, Springer, 1976, vii+201 pages | Zbl 0343.32002

[34] Arturo Giles Flores Specialization to the tangent cone and Whitney equisingularity, Bull. Soc. Math. Fr., Volume 141 (2013) no. 2, pp. 299-342 | Zbl 1286.14052

[35] Terence Gaffney Auréoles and integral closure of modules, Stratifications, singularities and differential equations. II: Stratifications and topology of singular spaces (Travaux en Cours) Volume 55, Hermann, 1990, pp. 55-62 | Zbl 0889.32034

[36] Terence Gaffney Integral closure of modules and Whitney equisingularity, Invent. Math., Volume 107 (1992) no. 2, pp. 301-322 | Zbl 0807.32024

[37] Terence Gaffney Equisingularity of Plane Sections, t 1 Condition, and the Integral Closure of Modules, Real and complex singularities (Pitman Research Notes in Mathematics Series) Volume 333, Longman, 1995, pp. 95-111 | Zbl 0844.32019

[38] Terence Gaffney Multiplicities and equisingularity of ICIS germs, Invent. Math., Volume 123 (1996) no. 2, pp. 209-220 | Zbl 0846.32024

[39] Terence Gaffney Polar methods, invariants of pairs of modules and equisingularity, Real and complex singularities (Contemporary Mathematics) Volume 354, American Mathematical Society, 2002, pp. 113-136 | Zbl 1072.32020

[40] Terence Gaffney Generalized Buchsbaum-Rim Multiplicities and a Theorem of Rees, Commun. Algebra, Volume 31 (2003) no. 8, pp. 3811-3828 | Zbl 1036.3208

[41] Terence Gaffney Bi-Lipschitz equivalence, integral closure and invariants, Real and complex singularities (London Mathematical Society Lecture Note Series) Volume 380, Cambridge University Press, 2010, pp. 125-137 | Zbl 1223.32018

[42] Terence Gaffney; Robert Gassler Segre Numbers and Hypersurface Singularities, J. Algebr. Geom., Volume 8 (1999) no. 4, pp. 695-736 | Zbl 0971.13021

[43] Terence Gaffney; Steven L. Kleiman Specialization of integral dependence for modules, Invent. Math., Volume 137 (1999) no. 3, pp. 541-574 | Zbl 0980.32009

[44] Mark Goresky; Robert MacPherson Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Volume 14, Springer, 1988, xiv+272 pages | Zbl 0639.14012

[45] Phillip A. Griffiths Complex differential and integral geometry and curvature integrals associated to singularities of complex analytic varieties, Duke Math. J., Volume 45 (1978) no. 3, pp. 427-512 | Zbl 0409.53048

[46] Abramo Hefez; Steven L. Kleiman Notes on the duality of projective varieties, Geometry today (Progress in Mathematics) Volume 60, Birkhäuser, 1984, pp. 143-183 | Zbl 0579.14047

[47] Achim Hennings Fibre dimension of the Nash transformation (2014) (https://arxiv.org/abs/1410.8449v1)

[48] Jean-Pierre-Georges Henry; Michel Merle Limites de normales, conditions de Whitney et éclatements d’Hironaka, Singularities (Proceedings of Symposia in Pure Mathematics) Volume 40, American Mathematical Society, 1981, pp. 575-584 | Zbl 0554.32010

[49] Jean-Pierre-Georges Henry; Michel Merle; Claude Sabbah Sur la condition de Thom stricte pour un morphisme analytique complexe, Ann. Sci. Éc. Norm. Supér., Volume 17 (1984), pp. 217-268 | Zbl 0551.32012

[50] Manfred Herrmann; Shin Ikeda; Ulrich Orbanz Equimultiplicity and Blowing up, Springer, 1988, xvii+629 pages | Zbl 0649.13011

[51] Heisuke Hironaka Normal cones in analytic Whitney Stratifications, Publ. Math., Inst. Hautes Étud. Sci., Volume 36 (1970), pp. 127-138 | Zbl 0219.57022

[52] Friedrich Hirzebruch Topological methods in Algebraic Geometry, Classics in Mathematics, Springer, 1978, ix+234 pages | Zbl 0843.14009

[53] Audun Holme On the dual of a smooth variety, Algebraic geometry (Lecture Notes in Mathematics) Volume 732, Springer, 1978, pp. 144-156 | Zbl 0441.14016

[54] Alfrederic Josse; Françoise Pène On caustics by reflection of algebraic surfaces, Adv. Geom., Volume 16 (2016) no. 4, pp. 437-464 | Zbl 1387.14087

[55] Masaki Kashiwara Index theorem for a maximally overdetermined system of linear differential equations, Proc. Japan Acad., Volume 49 (1973), p. 803-804 | Zbl 0305.35076

[56] Masaki Kashiwara Systems of microdifferential equations, Progress in Mathematics, Volume 34, Birkhäuser, 1983, xv+159 pages | Zbl 0521.58057

[57] Ludger Kaup; Burchard Kaup Holomorphic Functions of Several Variables, De Gruyter Studies in Mathematics, Volume 3, Walter de Gruyter, 1983, xv+349 pages | Zbl 0528.32001

[58] Steven L. Kleiman The transversality of a general translate, Compos. Math., Volume 28 (1974), pp. 287-297 | Zbl 0288.14014

[59] Steven L. Kleiman The enumerative theory of Singularities, Real and complex singularities, Sijthoff & Noordhoff International Publishers, 1976, pp. 297-396 | Zbl 0385.14018

[60] Steven L. Kleiman About the conormal scheme, Complete intersections (Lecture Notes in Mathematics) Volume 1092, Springer, 1984, pp. 161-197 | Zbl 0547.14031

[61] Steven L. Kleiman Tangency and duality, Proceedings of the 1984 Vancouver conference in algebraic geometry (CMS Conference Proceedings) Volume 6, American Mathematical Society, 1984, pp. 163-225 | Zbl 0601.14046

[62] Steven L. Kleiman A generalized Teissier-Plücker formula, Classification of algebraic varieties (Contemporary Mathematics) Volume 162, American Mathematical Society, 1992, pp. 249-260 | Zbl 0820.14039

[63] Remi Langevin Courbure et singularités complexes, Comment. Math. Helv., Volume 54 (1979), pp. 6-16 | Zbl 0429.32008

[64] Gérard Laumon Degré de la variété duale d’une hypersurface à singularités isolées, Bull. Soc. Math. Fr., Volume 104 (1976) no. 1, pp. 51-63 | Zbl 0343.14014

[65] Gérard Laumon Transformations canoniques et spécialisation pour les 𝒟-modules filtrés, Systèmes différentiels et singularités (Astérisque) Volume 130, Société Mathématique de France, 1985, pp. 56-129 | Zbl 0591.14012

[66] Dũng Tràng Lê Sur un critère d’équisingularité, C. R. Acad. Sci., Paris, Sér. A, Volume 272 (1971), pp. 138-140 | Zbl 0209.24402

[67] Dũng Tràng Lê Calcul du nombre de Milnor d’une singularité isolée d’intersection complète, Funkts. Anal. Prilozh., Volume 8 (1974) no. 2, pp. 45-49 | Zbl 0351.32007

[68] Dũng Tràng Lê; Chakravarthi P. Ramanujam The invariance of Milnor number implies the invariance of the topological type, Am. J. Math., Volume 98 (1976), pp. 67-78 | Zbl 0351.32009

[69] Dũng Tràng Lê; Bernard Teissier Sur la géométrie des surfaces complexes, I. Tangentes exceptionelles, Am. J. Math., Volume 101 (1979) no. 2, pp. 420-452 | Zbl 0427.32012

[70] Dũng Tràng Lê; Bernard Teissier Variétés polaires locales et classes de Chern des variétés singulières, Ann. Math., Volume 114 (1981), pp. 457-491 (erratum in Ann. Math. 115 (1982), p. 668) | Zbl 0488.32004

[71] Dũng Tràng Lê; Bernard Teissier Cycles évanescents, sections planes, et conditions de Whitney II, Singularities (Proceedings of Symposia in Pure Mathematics) Volume 40, American Mathematical Society, 1983, pp. 65-103 | Zbl 0532.32003

[72] Dũng Tràng Lê; Bernard Teissier Limites d’espaces tangents en géométrie analytique, Comment. Math. Helv., Volume 63 (1988) no. 4, pp. 540-578 | Zbl 0658.32010

[73] Monique Lejeune-Jalabert; Bernard Teissier Clôture intégrale des idéaux et équisingularité, Ann. Fac. Sci. Toulouse, Math., Volume 17 (2008) no. 4, pp. 781-859 | Zbl 1171.13005

[74] Joseph Lipman Equisingularity and simultaneous resolution of singularities, Resolution of Singularities: a research textbook in tribute to Oscar Zariski (Progress in Mathematics) Volume 181, Birkhäuser, 2000, pp. 485-505 | Zbl 0970.14011

[75] John N. Mather Notes on topological stability, Mimeographed notes, Harvard, 1970 (revised version published in Bull. Am. Math. Soc. 49 (2012), no. 4, p. 475–506) | Zbl 1260.57049

[76] John N. Mather Stratifications and Mappings, Dynamical Systems, Academic Press, 1973, pp. 195-232 | Zbl 0286.58003

[77] Yutaka Matsui; Kiyoshi Takeuchi Generalized Plücker-Teissier-Kleiman formulas for varieties with arbitrary dual defect, Real and complex singularities, World Scientific, 2007, pp. 248-270 | Zbl 1124.14009

[78] Hideyuki Matsumura Commutative Ring Theory, Cambridge University Press, 2000

[79] John W. Milnor Singular points of complex hypersurfaces, Annals of Mathematics Studies, Volume 61, Princeton University Press, 1968, 122 pages | Zbl 0184.48405

[80] Tadeusz Mostowski; E. Rannou Complexity of the Computation of the Canonical Whitney Stratification of an Algebraic Set in n , Applied algebra, algebraic algorithms and error-correcting codes (Lecture Notes in Computer Science) Volume 539 (1991), pp. 281-291 | Zbl 0773.32023

[81] Vicente Navarro Aznar On the Chern Classes and the Euler Characteristic for nonsingular Complete Intersections, Proc. Am. Math. Soc., Volume 78 (1980) no. 1, pp. 143-148 | Zbl 0473.14020

[82] Walter D. Neumann; Anne Pichon Lipschitz geometry of complex surfaces: analytic invariants (2012) (https://arxiv.org/abs/1211.4897)

[83] Juan José Nuño-Ballesteros; B. Oréfice-Okamoto; João Nivaldo Tomazella The vanishing Euler characteristic of an isolated determinantal singularity, Isr. J. Math., Volume 197 (2013), pp. 475-495 (erratum in Isr. J. Math., 224 (2018), p. 505-512) | Zbl 1277.32028

[84] Donal OʼShea Computing limits of tangent spaces: singularities, computation and pedagogy, Singularity theory, World Scientific, 1991, pp. 549-573 | Zbl 0947.14029

[85] Adam Parusiński; Laurenţiu Păunescu Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity, Adv. Math., Volume 309 (2017), pp. 254-305 | Zbl 1375.32048

[86] Frédéric Pham Singularités des systèmes différentiels de Gauss-Manin, Progress in Mathematics, Volume 2, Birkhäuser, 1979, x+339 pages | Zbl 0524.32015

[87] Ragni Piene Polar classes of singular varieties, Ann. Sci. Éc. Norm. Supér., Volume 11 (1978) no. 2, pp. 247-276 | Zbl 0401.14007

[88] Ragni Piene Polar varieties revisited, 2013 (slides for the Workshop on Computer Algebra and Polynomials in Linz, www.ricam.oeaw.ac.at/specsem/specsem2013/workshop3/slides/piene.pdf)

[89] Ragni Piene Polar varieties revisited, Computer algebra and polynomials. Applications of algebra and number theory (Lecture Notes in Computer Science) Volume 8942, Springer, 2015, pp. 139-150 | Zbl 06585204

[90] Julius Plücker Theorie der algebraischen Kurven, Bonn, 1839

[91] Jean-Victor Poncelet Traité des propriétés projectives des figures, ouvrage utile à ceux qui s’occupent des applications de la Géométrie descriptive et d’opérations géométriques sur le terrain. Tome second., Gauthier-Villars, 1866 (available at http://gallica.bnf.fr/ark:/12148/bpt6k5484980j)

[92] Patrick Popescu-Pampu What is the genus?, Lecture Notes in Mathematics, Volume 2162, Springer, 2016, xvii+184 pages | Zbl 1353.55001

[93] David Rees α-transforms of local rings and a theorem on multiplicities of ideals, Proc. Camb. Philos. Soc., Volume 57 (1961), pp. 8-17 | Zbl 0111.24803

[94] Reinhold Remmert Holomorphe und meromorphe abbildungen complexe raüme, Math. Ann., Volume 133 (1957), pp. 328-370 | Zbl 0079.10201

[95] Claude Sabbah Quelques remarques sur la géométrie des espaces conormaux, Systèmes différentiels et singularités (Astérisque) Volume 130, Société Mathématique de France, 1985, pp. 161-192 | Zbl 0598.32011

[96] Mohab Safey El Din; Éric Schost Polar varieties and computation of one point in each connected component of a smooth algebraic set, Proceeedings of the 2003 ISSAC (2003), pp. 224-231 | Zbl 1072.68693

[97] J. Edson Sampaio Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones, Sel. Math., New Ser., Volume 22 (2016) no. 2, pp. 553-559 | Zbl 1338.32008

[98] Marie-Hélène Schwartz Champs radiaux sur une stratification analytique, Travaux en Cours, Volume 39, Hermann, 1991, x+185 pages | Zbl 0727.57026

[99] Marie-Hélène Schwartz Classes de Chern des ensembles analytiques, Actualités Mathématiques, Hermann, 2000, 216 pages | Zbl 1095.57500

[100] John G. Semple Some investigations in the geometry of curve and surface elements, Proc. Lond. Math. Soc., Volume 4 (1954), pp. 24-49 | Zbl 0055.14505

[101] Ana Cannas da Silva Lectures on Symplectic Geometry, Lecture Notes in Mathematics, Volume 1764, Springer, 2001, xii+217 pages | Zbl 1016.53001

[102] Aaron Simis; Karen E. Smith; Bernd Ulrich An algebraic proof of Zak’s inequality for the dimension of the Gauss image, Math. Z., Volume 241 (2002) no. 4, pp. 871-881 | Zbl 1079.14060

[103] Jawad Snoussi Limites d’espaces tangents à une surface normale, Comment. Math. Helv., Volume 76 (2001), pp. 61-88 | Zbl 0990.32005

[104] Jawad Snoussi Linear components of the tangent cone in the Nash modification of a complex surface singularity, J. Singul., Volume 3 (2011), pp. 83-88 | Zbl 1292.32016

[105] Marcio G. Soares Holomorphic foliations and characteristic numbers, Commun. Contemp. Math., Volume 7 (2005) no. 5, pp. 583-596 | Zbl 1088.32020

[106] Mark Spivakovsky Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Ann. Math., Volume 131 (1990) no. 3, pp. 411-491 | Zbl 0719.14005

[107] Bernard Teissier Cycles évanescents, sections planes et conditions de Whitney, Singularites à Cargese (Astérisque) Volume 7-8, Société Mathématique de France, 1974, pp. 282-362 | Zbl 0295.14003

[108] Bernard Teissier Introduction to equisingularity problems, Algebraic Geometry Arcata 1974 (Proceedings of Symposia in Pure Mathematics) Volume 29, American Mathematical Society, 1974, pp. 593-632

[109] Bernard Teissier The hunting of invariants in the geometry of discriminants, Real and Complex singularities, Oslo 1976, Sijthoff & Noordhoff International Publishers, 1976, pp. 565-678 | Zbl 0388.32010

[110] Bernard Teissier Résolution simultanée, II, Séminaire sur les Singularités des Surfaces 1976-77 (Lecture Notes in Mathematics) Volume 777, Springer, 1976

[111] Bernard Teissier Variétés polaires 1: Invariants polaires des singularités d’hypersurfaces, Invent. Math., Volume 40 (1977) no. 3, pp. 267-292 | Zbl 0446.32002

[112] Bernard Teissier Variétés Polaires 2: Multiplicités polaires, sections planes, et conditions de Whitney, Actes de la conférence de géométrie algébrique à La Rábida 1981 (Lecture Notes in Mathematics) Volume 961, Springer, 1981, pp. 314-491 | Zbl 0585.14008

[113] Bernard Teissier Sur la classification des singularités des espaces analytiques complexes, Proceedings of the International Congress of Mathematicians, 1983, Warszawa, North-Holland, 1983, pp. 763-781 | Zbl 0574.32015

[114] Bernard Teissier Apparent contours from Monge to Todd, 1830–1930: A century of geometry (Paris, 1989) (Lecture Notes in Physics) Volume 402, Springer, 1992, pp. 55-62

[115] Evgueni A. Tevelev Projective duality and homogeneous spaces, Encyclopaedia of Mathematical Sciences, Volume 133, Springer, 2005, xiv+250 pages

[116] René Thom Ensembles et morphismes stratifiés, Bull. Am. Math. Soc., Volume 75 (1969), pp. 240-284 | Zbl 0197.20502

[117] Mihai Tibăr Limits of tangents and minimality of complex links, Topology, Volume 42 (2003) no. 3, pp. 629-639 | Zbl 1027.32028

[118] John A. Todd The arithmetical invariants of algebraic loci, Proc. Lond. Math. Soc., Volume 43 (1937), pp. 190-225

[119] John A. Todd The geometrical invariants of algebraic loci, Proc. Lond. Math. Soc., Volume 43 (1937), pp. 127-138 | Zbl 63.0624.01

[120] John A. Todd The geometrical invariants of algebraic loci II, Proc. Lond. Math. Soc., Volume 45 (1939), pp. 410-424 | Zbl 0061.32908

[121] Jean-Louis Verdier Stratifications de Whitney et Théorème de Bertini-Sard, Invent. Math., Volume 36 (1976), pp. 295-312 | Zbl 0333.32010

[122] Robert J. Walker Algebraic curves, Dover Publications, 1962, x+210 pages | Zbl 0103.38202

[123] Andrew H. Wallace Homology theory on algebraic varieties, International Series of Monographs on Pure and Applied Mathematics, Volume 6, Pergamon Press, 1958, viii+115 pages | Zbl 0100.16303

[124] Hassler Whitney Local properties of analytic varieties, Differential and Combinatorial Topology, Princeton University Press, 1965, pp. 205-244 | Zbl 0129.39402

[125] Hassler Whitney Tangents to an analytic variety, Ann. Math., Volume 81 (1965) no. 3, pp. 496-549 | Zbl 0152.27701

[126] Fyodor L. Zak Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, American Mathematical Society, 1993, vii+164 pages | Zbl 0795.14018

[127] Oscar Zariski Le problème des modules pour les branches planes, Hermann, 1986, vii+212 pages