Local polar varieties in the geometric study of singularities

Flores, Arturo Giles; Teissier, Bernard

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 $X\subset {ℂ}^n$ des hyperplans de ${ℂ}^n$ 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 $X\subset {ℂ}^n$ of hyperplanes of ${ℂ}^n$ 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[18] Briancon, Joël; Speder, Jean-Paul 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

[19] Briancon, Joël; Speder, Jean-Paul Les conditions de Whiney impliquent ${\mu }^{*}$ constant, Ann. Inst. Fourier, Volume 26 (1976) no. 2, pp. 153-163 | DOI | Zbl

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

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

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

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

[24] Comte, Georges 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 | MR | Zbl

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

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

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

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

[29] Ein, Lawrence The ramification divisor for branched coverings of ${\mathbf{P}}^n$, Math. Ann., Volume 261 (1982), pp. 483-485 | MR | Zbl

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

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

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

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

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

[35] Gaffney, Terence 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 | MR | Zbl

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

[37] Gaffney, Terence 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 | MR | Zbl

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

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

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

[41] Gaffney, Terence 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 | DOI | MR | Zbl

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

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

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

[45] Griffiths, Phillip A. 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 | MR | Zbl

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

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

[48] Henry, Jean-Pierre-Georges; Merle, Michel 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[70] Lê, Dũng Tràng; Teissier, Bernard 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 | DOI | Zbl

[71] Lê, Dũng Tràng; Teissier, Bernard 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 | DOI | Zbl

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

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

[74] Lipman, Joseph 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 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[83] Nuño-Ballesteros, Juan José; Oréfice-Okamoto, B.; Tomazella, João Nivaldo 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 | DOI | MR | Zbl

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

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

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

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

[88] Piene, Ragni 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) | Zbl

[89] Piene, Ragni 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 | MR | Zbl

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

[91] Poncelet, Jean-Victor 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] Popescu-Pampu, Patrick What is the genus?, Lecture Notes in Mathematics, 2162, Springer, 2016, xvii+184 pages | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[112] Teissier, Bernard 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

[113] Teissier, Bernard 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

[114] Teissier, Bernard 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 | MR

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

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

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

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

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

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

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

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

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

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

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

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

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