Real $J$-curves with deep nests on ruled surfaces

The paper is divided in two independent parts. The goal of the first part (§ 2) is to present a new proof of the complex orientation formula obtained by S. Yu. Orevkov's [9], which allows one to generalize this formula to $J$-curves with deep nests on ruled surfaces. In particular, this yields an analog of this formula for separating real algebraic curves in $\mathbb C P^2$ with two nests.
In the second part (§ 3), analogs of the inequalities of Arnol'd [1] and Rokhlin [11] are obtained for separating real $J$-curves with deep nests on ruled surfaces.