zbMATH — the first resource for mathematics

Projective \(M\)-cubics and \(M\)-quartics in general position with a maximally intersecting pair of ovals. (English. Russian original) Zbl 0992.14022
St. Petersbg. Math. J. 11, No. 5, 837-852 (2000); translation from Algebra Anal. 11, No. 5, 166-184 (2000).
Let \(C_n\) denote the set of points of an \(M\)-curve of degree \(n\) in the real projective plane \(\mathbb{R}\mathbb{P}^2\). The cubic \(M\)-curve \(C_3\) consists of two disjoint topological circles \(O_3\) and \(J_3\). The oval \(O_3\) splits \(\mathbb{R}\mathbb{P}^2\) and the odd branch \(J_3\) does not split it. The \(M\)-quartic \(C^4\) consists of four ovals lying outside one another. It is assumed along the paper that one of these ovals, denoted \(O_4\), meets \(O_3\) at 12 distinct real points. Consequently, the other three ovals of \(C_4\) and the odd branch \(J_3\) have no intersection points, and so they are called free. The main result of the paper under review is a topological classification of the triplets \((\mathbb{R}\mathbb{P}^2, C_3 \cup C_4,C_3)\) under the assumptions of transversality of the intersection, the maximality of the number of common real points and the freedom of \(J_3\) and the three ovals of \(C_4\) distinct of \(O_4\). The paper corrects a minor mistake in section VIIIa of a previous article by G. M. Polotovskij [in: Real algebraic geometry, Proc. Conf., Rennes/Fr. 1991, Lect. Notes Math. 1524, 52-74 (1992; Zbl 0789.14028)].

14P25 Topology of real algebraic varieties
57M25 Knots and links in the \(3\)-sphere (MSC2010)
14H45 Special algebraic curves and curves of low genus