Geodesic
Regular homotopy of Hurwitz curves
Izvestiya. Mathematics , Tome 68 (2004) no. 3, pp. 521-542

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We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or, more generally, two curves with $A$-type singularities) in the Hirzebruch surface $\boldsymbol F_N$ with the same homology classes and sets of singularities are regular homotopic. Moreover, they are symplectically regular homotopic if $C_0$ and $C_1$ are symplectic with respect to a compatible symplectic form. 
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     author = {Vik. S. Kulikov and D. Auroux and V. V. Shevchishin},
     title = {Regular homotopy of {Hurwitz} curves},
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     number = {3},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2004_68_3_a5/}
}
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Vik. S. Kulikov; D. Auroux; V. V. Shevchishin. Regular homotopy of Hurwitz curves. Izvestiya. Mathematics , Tome 68 (2004) no. 3, pp. 521-542. http://geodesic.mathdoc.fr/item/IM2_2004_68_3_a5/