On the disposition of cubic and pair of conics in a real projective plane

Abstract

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.