Publikacje

Abstract:

Let (ℓ, f) : (C2, 0) −> (C2, 0) be the germ of a holomorphic mapping such that ℓ = 0 is a smooth curve and f = 0 has an isolated singularity at 0 \in C2. We assume that ℓ = 0 is not a branch of f = 0. The direct image
of the critical locus of this mapping is called the discriminant curve. The role of Puiseux exponents of the branches of the discriminant is mysterious, and it is therefore of interest to determine when there is non-degeneracy. In this paper we describe the weighted initial forms of the discriminant curve with respect to its Newton diagram.Then we study the pairs (ℓ, f) for which the discriminant curve is non-degenerate in the Kouchnirenko sense

B I B L I O G R A F I A[1] E. Casas-Alvero, Local geometry of planar analytic morphisms, Asian J. Math., 11
(2007), 373–426.
[2] H. Eggers, Polarinvarianten und die Topologie von Kurvensingularitaten, Bonner
Mathematische Schriften 147 (1982).
[3] R. Ephraim, Special polars and curves with one place at infinity, Proc. Symp. Pure
Math., 40 (1983), 353–359.
[4] E. R. Garcia Barroso, Invariants des singularites de courbes planes et courbure des
fibres de Milnor, Doctoral thesis, La Laguna University, 1996. Servicio de Publicaciones
de la Universidad de La Laguna (2004), 216 pp.
[5] E. Garcia Barroso and J. Gwoździewicz, A discriminant criterion of irreducibility,
Kodai Math. J., 35 (2012), 403–414.
[6] J. Gwoździewicz, Invariance of the Jacobian Newton diagram, Math. Res. Lett., 19
(2012), 377–382.
[7] J. Gwoździewicz, Ephraim’s pencils, Int. Math. Res. Not. IMRN, 15 (2013), 3371–
3385. doi: 10.1093/imrn/rns148
[8] J. Gwoździewicz, A. Lenarcik and A. Ploski, Polar invariants of plane curve singularities:
intersection theoretical approach, Demonstratio Math., 43 (2010),
303–323.
[9] J. Gwoździewicz and A. Ploski, On the Merle formula for polar invariants, Bull. Soc.
Sci. Lett. Lódź, 41 (1991), 61–67.
[10] J. Gwoździewicz and A. Płoski, On the polar quotients of an analytic plane curve,
Kodai Math. J., 25 (2002), 43–53.
[11] S. Izumi, S. Koike and T. C. Kuo, Computations and stability of the Fukui invariant,
Compositio Math., 130 (2002), 49–73.
[12] A. G. Kouchnirenko, Polyedres de Newton et nombres de Milnor, Invent. Math., 32
(1976), 1–31.
[13] T. C. Kuo and Y. C. Lu, On analytic function germs of two complex variables, Topology,
16 (1977), 299–310.
[14] T. C. Kuo and A. Parusiński, Newton–Puiseux roots of Jacobian determinants, J. Algebraic
Geom., 13 (2004), 579–601.
[15] A. Lenarcik, M. Masternak and A. Płoski, Factorization of the polar curve and the
Newton polygon, Kodai Math. J., 26 (2003), 288–303.
[16] A. Lenarcik, Polar quotients of a plane curve and the Newton algorithm, Kodai
Math. J. 27 (2004), 336–353.
[17] A. Lenarcik, On the Lojasiewicz exponent, special direction and the maximal polar
quotient, arXiv:1112.5578.
[18] M. Merle, Invariants polaires des courbes planes, Invent. Math., 41 (1977), 103–111.
[19] F. Michel, Jacobian curves for normal complex surfaces, Contemp. Math., 475 (2008),
135–150.
[20] B. Teissier, Varietes polaires. I. Invariants polaires des singularites des hypersurfaces,
Invent. Math., 40 (1977), 267–292.
[21] B. Teissier, The hunting of invariants in the geometry of discriminants, in: Proc. Ninth
Nordic Summer School, Oslo, 1976 (1978), pp. 565–678.
[22] B. Teissier, Polyedre de Newton jacobien et equisingularite, Seminaire sur les Singularites,
Publ. Math. Univ. Paris VII, 7 (1980), 193–221. See also ArXiv: 1203.5595.
[23] R. J. Walker, Algebraic Curves, Princeton Mathematical Series, vol. 13. Princeton
University Press (Princeton, N.J., 1950), 201 pp.
[24] O. Zariski, Le probleme des modules pour les branches planes, Hermann (Paris, 1986),
212 pp.

POWRÓT

Strona główna > Publikacje

Non-degeneracy of the discriminant