Abstract
Let C = f(x,y) = 0 be a germ of a reduced plane curve. As examples of the basic invariants of a plane curve, we have the Milnor number, the number of irreducible components, the resolution complexity, the Puiseux pairs of the irreducible components and their intersection multiplicities. In fact, the Puiseux pairs of the irreducible components and their intersection multiplicities are enough to describe the embedding topological type of C (see [17], [9]).
This work is done when the author is visiting at Mathematisches Institut, University of Basel in 1991. He thanks to the Institut for their support.
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References
K. Brauner Klassifikation der Singularitäten algebroider Kurven. Abh. Math. Semin. Hamburg Univ. vol 6. 1928.
V.I. Danilov The geometry of toric varieties Russian Math. Surveys. Pages 97–154 vol 33:2, 1978.
P. Griffiths and J. Harris Principles of Algebraic Geometry. A Wiley-Interscience Publication. New York-Chichester-Brisbane-Toronto. 1978.
A.G. Khovanskii Newton polyhedra and toral varieties. Funkts. Anal. Prilozhen. vol 11, No. 4, pages 56–67 1977.
A.G. Khovanskii Newton polyhedra and the genus of complete intersections Funkts. Anal. Prilozhen. vol 12, No. 1. pages 51–61, 1977.
A.G. Kouchnirenko Polyèdres de Newton et Nombres de Milrior Inventiones Math, vol 32, pages 1–32, 1976.
G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat Toroidal Embeddings, Lecture Notes in Math. 339. Springer-Verlag. Berlin-Heidelberg-New York 1973.
H.B. Laufer Normal Two-Dimensional Singularities. Annals of Math. Studies. 71. Princeton Univ. Press. Princeton 1971.
D.T. Lê Sur un critére d’equismgularité. C.R.Acad.Sci.Paris, Ser. A–B. vol 272. pages 138–140 1971
D.T. Lê and M. Oka On the Resolution Complexity of Plane Curves Titech-Mathfg preprints series 10–93. 1993.
J. Milnor Singular Points of Complex Hypersurface. Annals Math. Studies vol 61 Princeton Univ. Press. Princeton 1968.
T. Oda Convex Bodies and Algebraic Geometry. Springer-Verlag. Berlin-Heidelberg-New York. 1987.
M. Oka On the Resolution of Hypersurface Singularities. Advanced Study in Pure Mathematics, vol 8, pages 405–436, 1986
M. Oka Principal zeta-function of non-degenerate complete intersection singularity J. Fac. Sci., Univ. of Tokyo vol 37, No. 1. pages 11–32. 1990.
M. Oka On the topology of full non-degenerate complete intersection variety Nagoya Math. J. vol 121. pages 137–148, 1991.
M. Oka Note on the resolution complexity of plane curve. Proceeding of the Workshop on Resolution of Singularities vol 1.82 (742) pages 125–148. 1993
O. Zariski Algebraic surfaces. Springer-Verlag. Berlin Heidelberg New York. 1934
O. Zariski Studies in equisingularity, I. Amer. J. Math, vol 31 pages 507–537. 1965.
O. Zariski Le Probleme des modules pour les branches planes. Cours donné au Centre de Mathematiqués de l’Ecole Polytechnique. 1973.
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© 1996 Birkhäuser Verlag Basel/Switzerland
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Oka, M. (1996). Geometry of Plane Curves via Toroidal Resolution. In: López, A.C., Macarro, L.N. (eds) Algebraic Geometry and Singularities. Progress in Mathematics, vol 134. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9020-5_5
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