 
Summary: REGULAR HOMOTOPY OF HURWITZ CURVES
DENIS AUROUX, VIKTOR S. KULIKOV,
AND VSEVOLOD V. SHEVCHISHIN
Abstract. We prove that any two irreducible cuspidal Hurwitz
curves C0 and C1 (or more generally, curves with Atype singu
larities) in the Hirzebruch surface F N with coinciding homology
classes and sets of singularities are regular homotopic; and symplec
tically regular homotopic if C0 and C1 are symplectic with respect
to a compatible symplectic form.
0. Introduction
In this paper, we deal with Jholomorphic curves in the projective
plane and Hurwitz curves (in particular, algebraic curves) in the Hirze
bruch surfaces F N which imitate the behavior of plane algebraic curves
with respect to pencils of lines (the definition of Hurwitz curves is given
in Section 2). We restrict ourselves to the case when Hurwitz curves
can have only singularities of the types An with n 0 (i.e., which
are locally given by y2
= xn+1
) and also socalled negative nodes (see
Section 2).
