Arrangements of an $M$-quintic with respect to a conic that maximally intersects its odd branch

Abstract

0.1. Statement of main results. Connected components of the set of real points of a plane projective real curve are called branches. A branch is called even (or an oval), if it is zero-homologous in RP. Otherwise it is called odd (or a pseudoline). Theorem 1. a). Let J be a tame almost complex structure in CP which is invariant under the complex conjugation, and let C5 and C2 be nonsingular real J holomorphic M -curves in RP of degrees 5 and 2 respectively. Let J5 be the odd branch of C5. Suppose that J5 intersects C2 at ten distinct real points. Then the arrangement of C5 ∪C2 in RP is one of those listed in Sect. 0.5 up to isotopy. All these arrangements are realizable. b). All the arrangements except the six of them labeled by ”∃/ alg.” or ”∃/ alg.” are realizable by real algebraic curves of degrees 5 and 2. c). The two arrangements labeled by ”∃/ alg.” are unrealizable by real algebraic curves of degrees 5 and 2.