Zariski’s dimensionality type of singularities. Case of dimensionality type 2
Authors:
Adam Parusiński and Laurenţiu Păunescu
Journal:
J. Algebraic Geom. 33 (2024), 117-142
DOI:
https://doi.org/10.1090/jag/815
Published electronically:
November 2, 2022
MathSciNet review:
4693575
Full-text PDF
Abstract |
References |
Additional Information
Abstract: In the 1970s O. Zariski introduced a general theory of equisingularity for algebroid and algebraic hypersurfaces over an algebraically closed field of characteristic zero. His theory builds up on understanding the dimensionality type of hypersurface singularities, notion defined recursively by considering the discriminants loci of successive “generic” corank $1$ projections. The theory of singularities of dimensionality type 1, that is the ones appearing generically in codimension 1, was developed by Zariski in his foundational papers on equisingular families of plane curve singularities. In this paper we completely settle the case of dimensionality type 2, by studying Zariski equisingular families of surfaces singularities, not necessarily isolated, in the three-dimensional space.
References
- Shreeram Abhyankar, On the ramification of algebraic functions, Amer. J. Math. 77 (1955), 575–592. MR 71851, DOI 10.2307/2372643
- Lev Birbrair, Walter D. Neumann, and Anne Pichon, The thick-thin decomposition and the bilipschitz classification of normal surface singularities, Acta Math. 212 (2014), no. 2, 199–256. MR 3207758, DOI 10.1007/s11511-014-0111-8
- J. Briançon and J. P. G. Henry, Équisingularité générique des familles de surfaces à singularité isolée, Bull. Soc. Math. France 108 (1980), no. 2, 259–281 (French, with English summary). MR 606093, DOI 10.24033/bsmf.1920
- Heisuke Hironaka, Normal cones in analytic Whitney stratifications, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 127–138. MR 277759, DOI 10.1007/BF02684601
- Heisuke Hironaka, On Zariski dimensionality type, Amer. J. Math. 101 (1979), no. 2, 384–419. MR 527999, DOI 10.2307/2373985
- Heinrich W. E. Jung, Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen $x,y$ in der Umgebung einer Stelle $x=a,\
y=b$, J. Reine Angew. Math. 133 (1908), 289–314 (German). MR 1580742, DOI 10.1515/crll.1908.133.289
- Joseph Lipman, Equisingularity and simultaneous resolution of singularities, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 485–505. MR 1748631, DOI 10.1007/978-3-0348-8399-3_{1}7
- Ignacio Luengo, An example concerning a question of Zariski, Bull. Soc. Math. France 113 (1985), no. 4, 379–386 (English, with French summary). MR 850774, DOI 10.24033/bsmf.2039
- Walter D. Neumann and Anne Pichon, Lipschitz geometry of complex curves, J. Singul. 10 (2014), 225–234. MR 3300297, DOI 10.5427/jsing.2014.10o
- A. Parusiński, Algebro-geometric equisingularity of Zariski, in Handbook of geometry and topology of singularities II, Springer, Cham, 2021, pp. 177–222.
- Adam Parusiński and Laurenţiu Păunescu, Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity, Adv. Math. 309 (2017), 254–305. MR 3607278, DOI 10.1016/j.aim.2017.01.016
- A. Parusiński and L. Păunescu, Lipschitz stratification of complex hypersurfaces in codimension 2, J. Eur. Math. Soc. (2022), published online first.
- Adam Parusiński and Guillaume Rond, The Abhyankar-Jung theorem, J. Algebra 365 (2012), 29–41. MR 2928451, DOI 10.1016/j.jalgebra.2012.05.003
- F. Pham and B. Teissier, Fractions lipschitziennes d’une algebre analytique complexe et saturation de Zariski, Prépublications École Polytechnique No. M17.0669 (1969). https://hal.archives-ouvertes.fr/hal-00384928.
- Bernard Teissier, The hunting of invariants in the geometry of discriminants, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff & Noordhoff, Alphen aan den Rijn, 1977, pp. 565–678. MR 568901
- Bernard Teissier, Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney, Algebraic geometry (La Rábida, 1981) Lecture Notes in Math., vol. 961, Springer, Berlin, 1982, pp. 314–491 (French). MR 708342, DOI 10.1007/BFb0071291
- A. N. Varchenko, Algebro-geometrical equisingularity and local topological classification of smooth mappings, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congr., Montreal, QC, 1975, pp. 427–431. MR 424805
- A. N. Varčenko, Local topological properties of analytic mappings, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 883–916 (Russian). MR 331421
- A. N. Varčenko, Local topological properties of differentiable mappings, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1037–1090 (Russian). MR 383453
- Hassler Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. MR 387634
- Oscar Zariski, Studies in equisingularity. I. Equivalent singularities of plane algebroid curves, Amer. J. Math. 87 (1965), 507–536. MR 177985, DOI 10.2307/2373019
- Oscar Zariski, Studies in equisingularity. II. Equisingularity in codimension $1$ (and characteristic zero), Amer. J. Math. 87 (1965), 972–1006. MR 191898, DOI 10.2307/2373257
- Oscar Zariski, Studies in equisingularity. III. Saturation of local rings and equisingularity, Amer. J. Math. 90 (1968), 961–1023. MR 237493, DOI 10.2307/2373492
- Oscar Zariski, Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481–491. MR 277533, DOI 10.1090/S0002-9904-1971-12729-5
- Oscar Zariski, On equimultiple subvarieties of algebroid hypersurfaces, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 1425–1426. MR 389894, DOI 10.1073/pnas.72.4.1425
- Oscar Zariski, Foundations of a general theory of equisingularity on $r$-dimensional algebroid and algebraic varieties, of embedding dimension $r+1$, Amer. J. Math. 101 (1979), no. 2, 453–514. MR 528001, DOI 10.2307/2373987
- Oscar Zariski, Addendum to my paper: “Foundations of a general theory of equisingularity on $r$-dimensional algebroid and algebraic varieties, of embedding dimension $r+1$” [Amer. J. Math. 101 (1979), no. 2, 453–514; MR 81m:14005], Amer. J. Math. 102 (1980), no. 3, 649–651. MR 573104, DOI 10.2307/2374117
References
- Shreeram Abhyankar, On the ramification of algebraic functions, Amer. J. Math. 77 (1955), 575–592. MR 71851, DOI 10.2307/2372643
- Lev Birbrair, Walter D. Neumann, and Anne Pichon, The thick-thin decomposition and the bilipschitz classification of normal surface singularities, Acta Math. 212 (2014), no. 2, 199–256. MR 3207758, DOI 10.1007/s11511-014-0111-8
- J. Briançon and J. P. G. Henry, Équisingularité générique des familles de surfaces à singularité isolée, Bull. Soc. Math. France 108 (1980), no. 2, 259–281 (French, with English summary). MR 606093
- Heisuke Hironaka, Normal cones in analytic Whitney stratifications, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 127–138. MR 277759
- Heisuke Hironaka, On Zariski dimensionality type, Amer. J. Math. 101 (1979), no. 2, 384–419. MR 527999, DOI 10.2307/2373985
- Heinrich W. E. Jung, Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen $x,y$ in der Umgebung einer Stelle $x=a,\ y=b$, J. Reine Angew. Math. 133 (1908), 289–314 (German). MR 1580742, DOI 10.1515/crll.1908.133.289
- Joseph Lipman, Equisingularity and simultaneous resolution of singularities, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 485–505. MR 1748631, DOI 10.1007/978-3-0348-8399-3_17
- Ignacio Luengo, An example concerning a question of Zariski, Bull. Soc. Math. France 113 (1985), no. 4, 379–386 (English, with French summary). MR 850774
- Walter D. Neumann and Anne Pichon, Lipschitz geometry of complex curves, J. Singul. 10 (2014), 225–234. MR 3300297, DOI 10.5427/jsing.2014.10o
- A. Parusiński, Algebro-geometric equisingularity of Zariski, in Handbook of geometry and topology of singularities II, Springer, Cham, 2021, pp. 177–222.
- Adam Parusiński and Laurenţiu Păunescu, Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity, Adv. Math. 309 (2017), 254–305. MR 3607278, DOI 10.1016/j.aim.2017.01.016
- A. Parusiński and L. Păunescu, Lipschitz stratification of complex hypersurfaces in codimension 2, J. Eur. Math. Soc. (2022), published online first.
- Adam Parusiński and Guillaume Rond, The Abhyankar-Jung theorem, J. Algebra 365 (2012), 29–41. MR 2928451, DOI 10.1016/j.jalgebra.2012.05.003
- F. Pham and B. Teissier, Fractions lipschitziennes d’une algebre analytique complexe et saturation de Zariski, Prépublications École Polytechnique No. M17.0669 (1969). https://hal.archives-ouvertes.fr/hal-00384928.
- Bernard Teissier, The hunting of invariants in the geometry of discriminants, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 565–678. MR 0568901
- Bernard Teissier, Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney, Algebraic geometry (La Rábida, 1981) Lecture Notes in Math., vol. 961, Springer, Berlin, 1982, pp. 314–491 (French). MR 708342, DOI 10.1007/BFb0071291
- A. N. Varchenko, Algebro-geometrical equisingularity and local topological classification of smooth mappings, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 427–431. MR 0424805
- A. N. Varčenko, Local topological properties of analytic mappings, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 883–916 (Russian). MR 0331421
- A. N. Varčenko, Local topological properties of differentiable mappings, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1037–1090 (Russian). MR 0383453
- Hassler Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0387634
- Oscar Zariski, Studies in equisingularity. I. Equivalent singularities of plane algebroid curves, Amer. J. Math. 87 (1965), 507–536. MR 177985, DOI 10.2307/2373019
- Oscar Zariski, Studies in equisingularity. II. Equisingularity in codimension $1$ (and characteristic zero), Amer. J. Math. 87 (1965), 972–1006. MR 191898, DOI 10.2307/2373257
- Oscar Zariski, Studies in equisingularity. III. Saturation of local rings and equisingularity, Amer. J. Math. 90 (1968), 961–1023. MR 237493, DOI 10.2307/2373492
- Oscar Zariski, Some open questions in the theory of singularities, Bull. Amer. Math. Soc. 77 (1971), 481–491. MR 277533, DOI 10.1090/S0002-9904-1971-12729-5
- Oscar Zariski, On equimultiple subvarieties of algebroid hypersurfaces, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 1425–1426. MR 389894, DOI 10.1073/pnas.72.4.1425
- Oscar Zariski, Foundations of a general theory of equisingularity on $r$-dimensional algebroid and algebraic varieties, of embedding dimension $r+1$, Amer. J. Math. 101 (1979), no. 2, 453–514. MR 528001, DOI 10.2307/2373987
- Oscar Zariski, Addendum to my paper: “Foundations of a general theory of equisingularity on $r$-dimensional algebroid and algebraic varieties, of embedding dimension $r+1$” [Amer. J. Math. 101 (1979), no. 2, 453–514; MR 81m:14005], Amer. J. Math. 102 (1980), no. 3, 649–651. MR 573104, DOI 10.2307/2374117
Additional Information
Adam Parusiński
Affiliation:
Université Côte d’Azur, CNRS, LJAD, UMR 7351, 06108 Nice, France
Email:
adam.parusinski@univ-cotedazur.fr
Laurenţiu Păunescu
Affiliation:
School of Mathematics and Statistics, The University of Sydney, Sydney, New South Wales, 2006, Australia
ORCID:
0000-0001-5796-276X
Email:
laurentiu.paunescu@sydney.edu.au
Received by editor(s):
July 7, 2021
Received by editor(s) in revised form:
May 16, 2022
Published electronically:
November 2, 2022
Additional Notes:
The first author is grateful for the support and hospitality of the Sydney Mathematical Research Institute (SMRI). This work was partially supported by ANR project LISA (ANR-17-CE40-0023-03).
Article copyright:
© Copyright 2022
University Press, Inc.
