On the number of connected components of the parabolic curve

Abstract

Abstract. We construct a polynomial of degree d in two variables whose Hessian curve has (d − 2) 2 connected components using Viro patchworking. In particular, this implies the existence of a smooth real algebraic surface of degree d in RP 3 whose parabolic curve is smooth and has d(d − 2) 2 connected components.

On the geometric structure of certain real algebraic surfaces

“…After multiplication by ω n+2 we obtain the desired quadratic form (5). Now, we shall prove that the equator is an integral curve of the fields Y k , k = 1, 2.…”

“…We denote by Y 1 , Y 2 the two fields of lines defined by the form (5). It is worth mentioning that the fields Y k , k = 1, 2, are not defined, in general, on the whole sphere.…”

“…where (1, v, ω). Suppose that p is a flat point of the form (5). So, the origin of the vω-plane is a singular point of the fields (8) since it is a zero of S (1, v, ω).…”

“…Conversely, let us suppose that the origin of the vω-plane is a singular point of the fields defined by (8). Thus, S(1, 0, 0) = 0, that is, the point p is a flat point of the form (5).…”

“…The configuration of these sets, invariant under the action of the affine group (or projective group) on 3-space, is the basic affine geometric structure of the surface. One of the goals in projective and affine differential geometry has been the study of this basic geometric structure for smooth and also for algebraic surfaces, see for example [2,3,5,12,19,25].…”