Geodesic
On rigid germs of finite morphisms of smooth surfaces
Sbornik. Mathematics, Tome 211 (2020) no. 10, pp. 1354-1381 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the article, we show that the germ of a finite morphism of smooth surfaces is rigid if and only if the germ of its branch curve has an $ADE$ singularity type. We establish a correspondence between the set of rigid germs of finite morphisms and the set of Belyi rational functions $f\in\overline{\mathbb Q}(z)$. Bibliography: 10 titles.
Keywords: rigid germs of finite morphisms, Belyi functions. 
@article{SM_2020_211_10_a0,
     author = {Vik. S. Kulikov},
     title = {On rigid germs of finite morphisms of smooth surfaces},
     journal = {Sbornik. Mathematics},
     pages = {1354--1381},
     year = {2020},
     volume = {211},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2020_211_10_a0/}
}
TY  - JOUR
AU  - Vik. S. Kulikov
TI  - On rigid germs of finite morphisms of smooth surfaces
JO  - Sbornik. Mathematics
PY  - 2020
SP  - 1354
EP  - 1381
VL  - 211
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_2020_211_10_a0/
LA  - en
ID  - SM_2020_211_10_a0
ER  - 
%0 Journal Article
%A Vik. S. Kulikov
%T On rigid germs of finite morphisms of smooth surfaces
%J Sbornik. Mathematics
%D 2020
%P 1354-1381
%V 211
%N 10
%U http://geodesic.mathdoc.fr/item/SM_2020_211_10_a0/
%G en
%F SM_2020_211_10_a0
Vik. S. Kulikov. On rigid germs of finite morphisms of smooth surfaces. Sbornik. Mathematics, Tome 211 (2020) no. 10, pp. 1354-1381. http://geodesic.mathdoc.fr/item/SM_2020_211_10_a0/

[1] V. I. Arnol'd, “Normal forms for functions near degenerate critical points, the Weyl groups of $A_k$, $D_k$, $E_k$ and Lagrangian singularities”, Funct. Anal. Appl., 6:4 (1972), 254–272 | DOI | MR | Zbl

[2] W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces, Ergeb. Math. Grenzgeb. (3), 4, Springer-Verlag, Berlin, 1984, x+304 pp. | DOI | MR | Zbl

[3] G. V. Belyi, “On Galois extensions of a maximal cyclotomic field”, Math. USSR-Izv., 14:2 (1980), 247–256 | DOI | MR | Zbl

[4] H. Grauert, R. Remmert, “Komplexe Räume”, Math. Ann., 136 (1958), 245–318 | DOI | MR | Zbl

[5] Vik. S. Kulikov, “Dualizing coverings of the plane”, Izv. Math., 79:5 (2015), 1013–1042 | DOI | DOI | MR | Zbl

[6] D. Mumford, “The topology of normal singularities of an algebraic surface and a criterion for simplicity”, Inst. Hautes Études Sci. Publ. Math., 9 (1961), 5–22 | DOI | MR | Zbl

[7] I. R. Shafarevich, Basic algebraic geometry, Grundlehren Math. Wiss., 213, Springer-Verlag, New York–Heidelberg, 1974, xv+439 pp. | MR | MR | Zbl | Zbl

[8] I. R. S̆afarevic̆, B. G. Averbuh, Ju. R. Vaĭnberg, A. B. Z̆iz̆c̆enko, Ju. I. Manin, B. G. Moĭs̆ezon, G. N. Tjurina, A. N. Tjurin, “Algebraic surfaces”, Proc. Steklov Inst. Math., 75 (1965), 1–281 | MR | MR | Zbl

[9] J. M. Wahl, “Equisingular deformations of plane algebroid curves”, Trans. Amer. Math. Soc., 193 (1974), 143–170 | DOI | MR | Zbl

[10] O. Zariski, “Studies in equisingularity. I. Equivalent singularities of plane algebroid curves”, Amer. J. Math., 87:2 (1965), 507–536 | DOI | MR | Zbl