Paper detail

Regular homotopy of Hurwitz curves

We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or more generally, curves with A-type singularities) in the Hirzebruch surface $F_N$ with coinciding homology classes and sets of singularities are regular homotopic; and symplectically regular homotopic if $C_0$ and $C_1$ are symplectic with respect to a compatible symplectic form.