Geodesic
Braid monodromy factorizations and diffeomorphism types
Izvestiya. Mathematics , Tome 64 (2000) no. 2, pp. 311-341

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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). 
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     author = {Vik. S. Kulikov and M. Teicher},
     title = {Braid monodromy factorizations and diffeomorphism types},
     journal = {Izvestiya. Mathematics },
     pages = {311--341},
     publisher = {mathdoc},
     volume = {64},
     number = {2},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2000_64_2_a3/}
}
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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/