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EMBED DINGS OF LINE IN THE PLANE AND ABHYANKAR-MOH EPIMORPHISM THEOREM
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 Title & Authors
EMBED DINGS OF LINE IN THE PLANE AND ABHYANKAR-MOH EPIMORPHISM THEOREM
Joe, Do-Sang; Park, Hyung-Ju;
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 Abstract
In this paper, we consider the parameter space of the rational plane curves with uni-branched singularity. We show that such a parameter space is decomposable into irreducible components which are rational varieties. Rational parametrizations of the irreducible components are given in a constructive way, by a repeated use of Abhyankar-Moh Epimorphism Theorem. We compute an enumerative invariant of this parameter space, and include explicit computational examples to recover some classically-known invariants.
 Keywords
embedding of affine line;Abhyankar-Moh epimorphism theorem;enumerative problems;Puiseux pairs;
 Language
English
 Cited by
1.
A POLAR, THE CLASS AND PLANE JACOBIAN CONJECTURE,;

대한수학회보, 2010. vol.47. 1, pp.211-219 crossref(new window)
 References
1.
S. Abhyankar and T. Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166.

2.
P. Aluffi, The enumerative geometry of plane cubics. II. Nodal and cuspidal cubics, Math. Ann. 289 (1991), no. 4, 543–572. crossref(new window)

3.
E. Brieskorn and H. Knorrer, Plane Algebraic Curves, Birkhauser Verlag, Basel, 1986.

4.
H. Flenner and M. Zaidenberg, On a class of rational cuspidal plane curves, Manuscripta Math. 89 (1996), no. 4, 439–459. crossref(new window)

5.
W. Fulton, Intersection Theory, Springer-Verlag, Berlin, 1984.

6.
M. H. Gizatullin and V. I. Danilov, Automorphisms of affine surfaces. I, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 3, 523–565, 703.

7.
I. M. Gel'fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Mathematics: Theory & Applications. Birkhauser, 1994.

8.
C. G. Gibson, Elementary Geometry of Algebraic Curves: An Undergraduate Introduction, Cambridge University Press, Cambridge, 1998.

9.
G.-M. Greuel, C. Lossen, and E. Shustin, Introduction to Singularities and Deformations, Springer, 2007.

10.
G.-M. Greuel, G. Pfister, and H. Schonemann, Singular 3.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2005. http://www.singular.uni-kl.de.

11.
M. Kazaryan, Multisingularities, cobordisms, and enumerative geometry, Russian Math. Surveys 58 (2003), no. 4, 665–724 crossref(new window)

12.
M. Kazaryan, Characteristic Classes in Singularity Theory, Doctoral Dissertation (habilitation thesis), Steklov Math. Inst., 2003.

13.
D. Kerner, Enumeration of unisingular algebraic curves, Math. AG/0407358, 2004.

14.
S. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297.

15.
S. Kleiman and R. Speiser, Enumerative geometry of cuspidal plane cubics, Proceedings of the 1984 Vancouver conference in algebraic geometry, 227–268, CMS Conf. Proc., 6, Amer. Math. Soc., Providence, RI, 1986.

16.
A. G. Kouchnirenko, Polyedres de newton et nombres de milnor, Invent. Math. 32 (1976), no. 1, 1–31. crossref(new window)

17.
A. van den Essen, Polynomial Automorphisms, Birkhauser, Basel, 2000.

18.
I. Vainsencher, Counting divisors with prescribed singularities, Trans. Amer. Math. Soc. 267 (1981), no. 2, 399–422. crossref(new window)

19.
A. van den Essen and J. Yu, The D-resultant, singularities and the degree of unfaithfulness, Proc. Amer. Math. Soc. 125 (1997), no. 3, 689–695. crossref(new window)

20.
H. Yoshihara, On plane rational curves, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 4, 152–155. crossref(new window)