Rigid isotopies of real trigonal curves on the Hirzebruch surfaces

A rigid isotopy of nonsingular real algebraic curves on a scroll is a path in the space of such curves of a given bidegree. For real algebraic curves of bidegree $(m,3)$ on the Hirzebruch surface $\Sigma_1$ (the projective plane with a point blown up) we obtain the rigid isotopy classification of nonsingular curves and give some corollaries for the space of curves with a single node or a cusp on a hyperboloid and on $\Sigma_2$.