A combinatorial approach to singularities of normal surfaces
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 3, pp. 461-491.

In this paper we study generic coverings of ℂ 2 branched over a curve s.t. the total space is a normal analytic surface, in terms of a graph representing the monodromy of the covering, called monodromy graph. A complete description of the monodromy graphs and of the local fundamental groups is found in case the branch curve is {x n =y m } (with n≤m) and the degree of the cover is equal to n or n-1.

Classification : 32S25, 32S05 
@article{ASNSP_2003_5_2_3_461_0,
     author = {Manfredini, Sandro},
     title = {A combinatorial approach to singularities of normal surfaces},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {461--491},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {3},
     year = {2003},
     mrnumber = {2020857},
     zbl = {1170.32313},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2003_5_2_3_461_0/}
}
TY  - JOUR
AU  - Manfredini, Sandro
TI  - A combinatorial approach to singularities of normal surfaces
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2003
SP  - 461
EP  - 491
VL  - 2
IS  - 3
PB  - Scuola normale superiore
UR  - http://archive.numdam.org/item/ASNSP_2003_5_2_3_461_0/
LA  - en
ID  - ASNSP_2003_5_2_3_461_0
ER  - 
%0 Journal Article
%A Manfredini, Sandro
%T A combinatorial approach to singularities of normal surfaces
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2003
%P 461-491
%V 2
%N 3
%I Scuola normale superiore
%U http://archive.numdam.org/item/ASNSP_2003_5_2_3_461_0/
%G en
%F ASNSP_2003_5_2_3_461_0
Manfredini, Sandro. A combinatorial approach to singularities of normal surfaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 3, pp. 461-491. http://archive.numdam.org/item/ASNSP_2003_5_2_3_461_0/

[Ar] E. Artin, Theory of braids, Ann. of Math. 48 (1947), 101-126. | MR | Zbl

[Bi] J. Birman, “Braids, links and mapping class groups”, Princeton University Press, 1975. | MR | Zbl

[BK] E. Brieskorn - H. Knörrer, “Plane algebraic curves”, Birkhäuser, 1986. | MR | Zbl

[Fi] G. Fischer, “Complex analytic geometry”, Lect. Notes in Math., Springer-Verlag, 1976. | MR | Zbl

[GrRe] H. Grauert - R. Remmert, Komplexe Räume, Math. Ann. 136 (1958), 245-318. | MR | Zbl

[GR] R. C. Gunning - H. Rossi, Analytic functions of several complex variables, Prentice-Hall series in modern analysis, (1965). | MR | Zbl

[La] H. Laufer, Normal two-dimensional singularities, Ann. of Math. Studies 71 Princeton Univ. Press, (1971). | MR | Zbl

[MKS] W. Magnus - A. Karrass - D. Solitar, “Combinatorial group theory”, Interscience, John Wiley and Sons, 1966. | MR | Zbl

[MP] S. Manfredini - R. Pignatelli, Generic covers branched over {x n =y m }, Topology Appl. 103 (2000), 1-31. | MR | Zbl

[Mo] B. G. Moishezon, Stable branch curves and braid monodromy, Lect. Notes in Math. 862 (1981), 107-192. | MR | Zbl

[Mu] D. Mumford, The topology of normal singularities and a criterion for simplicity, Inst. Hautes Études Sci., Publ. Math. 9 (1961), 5-22. | Numdam | MR | Zbl

[Na] R. Narasimhan, “Introduction to the theory of analytic spaces”, Lect. Notes in Math., Vol. 25, Springer-Verlag, 1966. | MR | Zbl

[O] M. Oka, On the fundamental group of the complement of certain plane curves, J. Math. Soc. Japan 30 (1978), 579-597. | MR | Zbl

[Pi] R. Pignatelli, Singolarità di superfici algebriche, Tesi di laurea, Università di Pisa (1994).

[T1] G. Teodosiu, A class of analytic coverings ramified over u 3 =v 2 , J. London Math. Soc. (2) 38 (1988), 231-242. | MR | Zbl

[T2] G. Teodosiu, Some analytic dihedral coverings, Stud. Cerc. Mat. 40 (1988), 155-159. | MR | Zbl

[VK] E. R. Van Kampen, On the fundamental group of an algebraic curve, Amer. J. Math. 55 (1933), 255-260. | JFM | MR

[Z] O. Zariski, On the problem of existence of algebraic functions of two variables possessing a given branch curve, Amer. J. Math. 51 (1929), 305-328. | JFM | MR