  The subject of this paper is the problem of arrangement of real algebraic
curves on real algebraic surfaces. In this paper we extend Rokhlin,
Kharlamov-Gudkov-Krakhnov and Kharlamov-Marin congruences for curves on
surfaces and give some applications of this extension.
  For some pairs consisting of a surface and a curve on this surface (in
particular for M-pairs) we introduce a new structure --- the complex separation
that is separation of the complement of curve into two surfaces. In accordance
with Rokhlin terminology the complex separation is a complex topological
characteristic of real algebraic varieties. The complex separation is similar
to complex orientations introduced by O.Ya.Viro (to the absolute complex
orientation in the case when a curve is empty and to the relative complex
orientation otherwise). In some cases we calculate the complex separation of a
surface (for example in the case when surface is the double branched covering
of another surface along a curve). With the help of these calculations
applications of the extension of Rokhlin congruence gives some new restrictions
for complex orientations of curves on a hyperboloid.
