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).