Arrangements of an M-quintic with Respect to a Conic That Maximally Intersects Its Odd Branch

Under certain assumptions, the arrangements mentioned in the title are classified up to isotopy. Their algebraic realizability is discussed. §0. Introduction 0.1. Statement of main results. The connected components of the set of real points of a plane projective real curve are called branches. A branch is even (or an oval) if it is zero-homologous in RP. Otherwise it is odd (or a pseudoline). Theorem 1. a) Let J be a tame almost complex structure in CP that is invariant under 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 Subsection 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. Theorem 2. a) Let J be a tame almost complex structure in CP that is invariant under complex conjugation, and let C5, L1, and L2 be nonsingular real J -holomorphic M -curves in RP of degrees 5, 1, and 1, respectively. Suppose that the odd branch J5 of C5 intersects each of the lines L1 and L2 at five distinct real points. Then either the arrangement of C5 ∪L1 ∪ L2 in RP is one of those listed in Subsections 0.6, 0.7 and in Figures 16.1–16.22, or C5 ∪ L1 ∪ L2 realizes one of the sixteen arrangements such that L1 = {x = 0} and L2 = {x + εy = 0}, where L1 is a line intersecting J5 at five points and 0 < ε 1. All these arrangements are realizable. b) All of them are algebraically realizable except the arrangement in Figure 16.12 and the five arrangements in Subsection 0.7 labeled by “ ∗ alg.” or “ alg.”. c) The three arrangements in Subsection 0.7 labeled by “ alg.” and that in Figure 16.12 are unrealizable by real algebraic curves. Theorems 1 and 2 are proved in §§2–7. A general outline of the proof is given in Subsections 0.2 and 0.4. Remark 1. I know a proof of the algebraic unrealizability of the four arrangements in Subsection 0.5 and of the two in Subsection 0.7 that are labeled by “ ∗ alg.”, but this paper is so long that I decided not to include it. Maybe, I shall write it somewhere else. 2000 Mathematics Subject Classification. Primary 57R52, 57R19.