Geodesic
Braid monodromy factorizations and diffeomorphism types
Izvestiya. Mathematics, Tome 64 (2000) no. 2, pp. 311-341 
Vik. S. Kulikov; M. Teicher. Braid monodromy factorizations and diffeomorphism types. Izvestiya. Mathematics, Tome 64 (2000) no. 2, pp. 311-341. http://geodesic.mathdoc.fr/item/IM2_2000_64_2_a3/
@article{IM2_2000_64_2_a3,
     author = {Vik. S. Kulikov and M. Teicher},
     title = {Braid monodromy factorizations and diffeomorphism types},
     journal = {Izvestiya. Mathematics},
     pages = {311--341},
     year = {2000},
     volume = {64},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2000_64_2_a3/}
}
TY  - JOUR
AU  - Vik. S. Kulikov
AU  - M. Teicher
TI  - Braid monodromy factorizations and diffeomorphism types
JO  - Izvestiya. Mathematics
PY  - 2000
SP  - 311
EP  - 341
VL  - 64
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/IM2_2000_64_2_a3/
LA  - en
ID  - IM2_2000_64_2_a3
ER  - 
%0 Journal Article
%A Vik. S. Kulikov
%A M. Teicher
%T Braid monodromy factorizations and diffeomorphism types
%J Izvestiya. Mathematics
%D 2000
%P 311-341
%V 64
%N 2
%U http://geodesic.mathdoc.fr/item/IM2_2000_64_2_a3/
%G en
%F IM2_2000_64_2_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we prove that if two cuspidal plane curves $B_1$ and $B_2$ have equivalent braid monodromy factorizations, then $B_1$ and $B_2$ are smoothly isotopic in $\mathbb C\mathbb P^2$. As a consequence, we obtain that if $S_1$, $S_2$ are surfaces of general type embedded in a projective space by means of a multiple canonical class and if the discriminant curves (the branch curves) $B_1$, $B_2$ of some smooth projections of $S_1$, $S_2$ to $\mathbb{CP}^2$ have equivalent braid monodromy factorizations, then $S_1$ and $S_2$ are diffeomorphic (as real four-dimensional manifolds).

[1] Artin E., “Theory of braids”, Ann. Math., 48 (1947), 101–126 | DOI | MR | Zbl

[2] Birman J., Braids, Links and Mapping Class Groups, Princeton University Press, 1975 | MR | Zbl

[3] Kulikov Vik. S., “O gipoteze Kizini”, Izv. RAN. Ser. matem., 63:6 (1999), 84–116 | MR

[4] Kulikov Vik. S., “Geometricheskaya realizatsiya $C$-grupp”, Izv. RAN. Ser. matem., 58:4 (1994), 194–203 | Zbl

[5] Kulikov Vik. S., “On the fundamental group of the complement of a hypersurface in ${\mathbb C}^n$”, Springer L.N.M., 1479, 1991, 122–130 | MR | Zbl

[6] Moishezon B., Teicher M., Braid Groups, Singularities and Algebraic Surfaces, Birkhauser (to appear)

[7] Moishezon B., Teicher M., “Braid group techniques in complex geometry. I: Line arrangements in $\mathbb C\mathbb P ^2$”, Contemporary Math., 78 (1988), 425–555 | MR | Zbl

[8] Moishezon B., Teicher M., “Braid group techniques in complex geometry. V: The fundamental group of a complement of a branch curve of a Veronese generic projection”, Communications in Analysis and Geometry, 4:11 (1996), 1–120 | MR | Zbl