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Transactions of the American Mathematical Society

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Analytic order of singular and critical points

Author: Eugenii Shustin
Journal: Trans. Amer. Math. Soc. 356 (2004), 953-985
MSC (2000): Primary 14F17, 14H20; Secondary 58K05
Published electronically: August 21, 2003
MathSciNet review: 1984463
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Abstract: We deal with the following closely related problems: (i) For a germ of a reduced plane analytic curve, what is the minimal degree of an algebraic curve with a singular point analytically equivalent (isomorphic) to the given one? (ii) For a germ of a holomorphic function in two variables with an isolated critical point, what is the minimal degree of a polynomial, equivalent to the given function up to a local holomorphic coordinate change? Classically known estimates for such a degree $d$ in these questions are $\sqrt{\mu}+1\le d\le \mu+1$, where $\mu$ is the Milnor number. Our result in both the problems is $d\le a\sqrt{\mu}$ with an absolute constant $a$. As a corollary, we obtain asymptotically proper sufficient conditions for the existence of algebraic curves with prescribed singularities on smooth algebraic surfaces.

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  • 1. J. Alexander and A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math. 140 (2000), no. 2, 303-325. MR 2001i:14024
  • 2. C. Ciliberto and R. Miranda, The Segre and Harbourne-Hirschowitz conjectures, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), NATO Sci. Ser. II Math. Phys. Chem., vol. 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 37-51. MR 2002k:14011
  • 3. E. D. Davis, $0$-dimensional subschemes of $\mathbb{P} ^2$: New applications of Castelnuovo's function, Ann. Univ. Ferrara 32 (1986), 93-107. MR 89c:14009
  • 4. S. Diaz and J. Harris, Ideals associated to deformations of singular plane curves, Trans. Amer. Math. Soc. 309 (1988), 433-468. MR 89m:14003
  • 5. G.-M. Greuel and C. Lossen, Equianalytic and equisingular families of curves on surfaces, Manuscripta Math. 91 (1996), no. 3, 323-342.MR 98g:14023
  • 6. G.-M. Greuel, C. Lossen, and E. Shustin, Plane curves of minimal degree with prescribed singularities, Invent. Math. 133 (1998), no. 3, 539-580. MR 99g:14035
  • 7. G.-M. Greuel, C. Lossen, and E. Shustin, Castelnuovo function, zero-dimensional schemes and singular plane curves, J. Alg. Geom. 9 (2000), no. 4, 663-710. MR 2001g:14045
  • 8. D. A. Gudkov and E. I. Shustin, On the intersection of the close algebraic curves, Topology (Leningrad, 1982), Lect. Notes Math., vol. 1060, Springer, Berlin, 1984, pp. 278-289. MR 86i:14008
  • 9. S. M. Gusein-Zade and N. N. Nekhoroshev, On $A_k$-singularity on plane curves of fixed degree, Func. Anal. i Prilozhen. 34 (2000), no. 3, 69-70 (Russian) (English translation in Func. Anal. Appl. 34 (2000), 214-215). MR 2001k:14055
  • 10. A. Hirano, Constructions of plane curves with cusps, Saitama Math. J. 10 (1992), 21-24. MR 94a:14029
  • 11. F. Hirzebruch, Singularities of algebraic surfaces and characteristic numbers, Contemp. Math. 58 (1986), 141-155. MR 87j:14057
  • 12. A. Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques, J. Reine Angew. Math. 397 (1989), 208-213. MR 90g:14021
  • 13. T. Keilen and I. Tyomkin, Existence of curves with prescribed topological singularities, Trans. Amer. Math. Soc. 354 (2002), 1837-1860. MR 2003a:14041
  • 14. C. Lossen, New asymptotics for the existence of plane curves with prescribed singularities, Comm. Algebra 27 (1999), 3263-3282.MR 2000e:14037
  • 15. J. N. Mather, Stability of $C^{\infty}$-mappings, III: Finitely determined map-germs, Publ. Math. IHES 35 (1968), 127-156. MR 43:1215a
  • 16. J. N. Mather and S.-T. Yau, Classification of isolated hypersurface singularities by their moduli algebras, Invent. Math. 69 (1982), 243-251. MR 84c:32007
  • 17. A. A. du Plessis and C. T. C. Wall, Singular hypersurfaces, versality and Gorenstein algebras, J. Alg. Geom 9 (2000), no. 2, 309-322. MR 2000k:14034
  • 18. F. Severi, Vorlesungen über Algebraische Geometrie (Anhang F), Leipzig, Teubner, 1921.
  • 19. E. Shustin, Real plane algebraic curves with prescribed singularities, Topology 32 (1993), 845-856. MR 95f:14049
  • 20. E. Shustin, Geometry of equisingular families of plane algebraic curves, J. Alg. Geom. 5 (1996), no. 2, 209-234. MR 97g:14025
  • 21. E. Shustin, Lower deformations of isolated hypersurface singularities, Algebra i Analiz 11 (1999), no. 5, 221-249 (English translation in St. Petersburg Math. J. 11 (2000), no. 5, 883-908). MR 2000m:32039
  • 22. B. L. van der Waerden, Einführung in die algebraische Geometrie, 2nd edition, Springer, Berlin, 1973. MR 49:8984
  • 23. J. Wahl, Equisingular deformations of plane algebroid curves, Trans. Amer. Math. Soc. 193 (1974), 143-170. MR 54:7460
  • 24. G. Xu, Curves in $\mathbb{P} ^2$ and symplectic packings, Math. Ann. 299 (1994), 609-613. MR 95f:14058
  • 25. G. Xu, Ample line bundles on smooth surfaces, J. Reine Angew. Math. 469 (1995), 199-209. MR 96k:14003

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Additional Information

Eugenii Shustin
Affiliation: School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel

Received by editor(s): July 5, 2002
Published electronically: August 21, 2003
Additional Notes: The author was partially supported by Grant No. G-616-15.6/99 of the German-Israeli Foundation for Research and Development and by the Hermann-Minkowski Minerva Center for Geometry at Tel Aviv University. This work was completed during the author’s RiP stay at the Mathematisches Forschunsinstitut Oberwolfach.
Article copyright: © Copyright 2003 American Mathematical Society

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