Parabolic lines on surfaces and gradient maps D.Panov, summary of the talk at Arnold's seminar 16.09.97 A surface in 3-space (affine, projective) contains points of 3 types: elliptic, hyperblic and parabolic. In the vicinity of an elliptic point the surface is locally convex and looks like an ellipsoid, in the vicinity of a hyperbolic point it looks like a one-sheeted hyperboloid. Points that are neither elliptic nor hyperbolic, are called parabolic. On a generic surface parabolic points form smooth lines. A hyperplane in the projective 3-space is not generic and consists entirely of parabolic points. After a generic deformation of the plane only a line of parabolic points remains. It is known that for the deformations in the class of cubic projective surfaces the parabolic line consists of four connected components. We show that there exist deformations for which the parabolic line consists of one component. Apart from this, the talk will contain a discussion of: -- double periodic functions on the two-dimensional torus, -- closed trajectories of the field of crosses (asymptotic lines) in hyperbolic regions, -- intersection indices related with the fields of crosses (and, more generally, fields of quadratic forms).