Affine plane curves with one place at infinity
Suzuki, Masakazu
Annales de l'Institut Fourier, Tome 49 (1999), p. 375-404 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Dans cet article on donne une nouvelle démonstration algébro-géométrique pour le théorème du semi-groupe d’Abhyankar-Moh sur les courbes planes affines avec un point à l’infini et le théorème réciproque dû à Sathaye-Stenerson. Les relations entre les divers invariants de ces courbes sont aussi expliquées géométriquement. Notre nouvelle démonstration donne un algorithme pour classifier les courbes planes affines avec un point à l’infini et un genre donné par ordinateur.

In this paper we give a new algebro-geometric proof to the semi-group theorem due to Abhyankar-Moh for the affine plane curves with one place at infinity and its inverse theorem due to Sathaye-Stenerson. The relations between various invariants of these curves are also explained geometrically. Our new proof gives an algorithm to classify the affine plane curves with one place at infinity with given genus by computer.

@article{AIF_1999__49_2_375_0,
     author = {Suzuki, Masakazu},
     title = {Affine plane curves with one place at infinity},
     journal = {Annales de l'Institut Fourier},
     volume = {49},
     year = {1999},
     pages = {375-404},
     doi = {10.5802/aif.1678},
     mrnumber = {2000f:14096},
     zbl = {0921.14017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1999__49_2_375_0}
}

Suzuki, Masakazu. Affine plane curves with one place at infinity. Annales de l'Institut Fourier, Tome 49 (1999) pp. 375-404. doi : 10.5802/aif.1678. http://gdmltest.u-ga.fr/item/AIF_1999__49_2_375_0/

[1] S.S. Abhyankar and T.T. Moh, Embeddings of the line in the plane, J. reine angev. Math., 276 (1975), 148-166. | MR 52 #407 | Zbl 0332.14004

[2] S.S. Abhyankar and T.T. Moh, On the semigroup of a meromorphic curve, Proc. Int. Symp. Algebraic Geometry, Kyoto (1977), 249-414. | Zbl 0408.14010

[3] N.A.' Campo and M.Oka, Geometry of plane curves via Tschirnhausen resolution tower, Osaka J. Math., Vol.33, No.4 (1996), 1003-1034. | MR 98d:14038 | Zbl 0904.14014

[4] M. Furushima, Finite groups of polynomial automorphisms in the complex affine plane(I), Mem. Fac. Sci. Kyushu Univ., 36 (1982), 85-105. | MR 84m:14016 | Zbl 0547.32015

[5] M. Miyanishi, Minimization of the embeddings of the curves into the affine plane, J. Kyoto Univ., Vol.36, No.2 (1996), 311-329. | MR 97j:14015 | Zbl 0908.14001

[6] J.A. Morrow, Compactifications of C2, Bull. Amer. Math. Soc., 78 (1972), 813-816. | MR 45 #8875 | Zbl 0257.32013

[7] Y. Nakazawa and M. Oka, Smooth plane curves with one place at infinity, to appear in J. Math. Soc. Japan, Vol.49, No.4 (1997). | MR 98i:14032 | Zbl 0903.14007

[8] W.D. Neumann, Complex algebraic curves via their links at infinity, Invent. Math., 98 (1989), 445-489. | MR 91c:57014 | Zbl 0734.57011

[9] C.P. Ramanujam, A topological characterization of the affine plane as an algebraic variety, Ann. Math., (2), 94 (1971), 69-88. | MR 44 #4010 | Zbl 0218.14021

[10] A. Sathaye and J. Stenerson, On plane polynomial curves, Algebraic geometry and its applications, C.L. Bajaj, ed., Springer, (1994), 121-142. | MR 95a:14032 | Zbl 0809.14022

[11] M. Suzuki, Propriété topologique des polynômes de deux variables complexes et automorphismes algébriques de l'espace C2, J. Math. Sci. Japan, 26 (1974), 241-257. | MR 49 #3188 | Zbl 0276.14001

[12] Zaidenberg and Lin, An irreducible simply connected algebraic curve in ℂ2 is equivalent to a quasihomogeneous curve, Soviet Math. Dokl., Vol.28, No.1 (1983), 200-204. | MR 85i:14018 | Zbl 0564.14014