Arrangements of an M-quintic with respect to a conic which maximally intersects its odd branch. Abstract

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

by S.Yu. Orevkov

Abstract:

We complete the classification up to isotopy of pseudoholomorphic arrangements of a nonsingular M-quintic and any (maybe reducible) conic on RP² under the condition that the curves are transvere to each other and all the intersection points lye on the same real branch of the quintic. Among such arrangements, there are 12 which are algebraically unrealizable, but here we prove this fact only for 6 of them. We realize algebraically all the other arrangements (except these 12 ones).

We prove also the following result about M-curves on RP² of any degree.

Theorem 3.
Let $C_2$ and $C_n$ be real pseudoholomorphic (for example, real algebraic) nonsingular M-curves of degrees 2 and n respectively. Suppose that $C_2$ intersects a branch $B_n$ of $C_n$ at 2n distinct points and all other branches of $C_n$ are contained in the exterior of $C_2$. Suppose that $B_n$ has the parity of n (i.e., either n is even, or n is odd and then $B_n$ is the odd branch of $C_n$). Then the intersection of $C_n$ with the interior of $C_2$ is the union of n parallel arcs.

In Appendix A, we give a new proof (very short) of a result originally proved in [1] that there does not exist an algebraic M-smoothing of four tangent branches of the type A(4,1,4).

In Appendix B, we give an example of a smooth algebraic curve of bidegree (3,6) on the quadratic cone which has an unremovable S-like zigzag (S-like zigzags are always removable in the class of real pseudoholomorphic curves).

In Appendix C, we prove that the isotopy types 1 U 1<1> U 1<18> and 1 U 1<7> U 1<12> are unrealizable by real algebraic curves of bidegree (4,16) on the Hirzebruch surface $F_4$. These isotopy types are realizable by real J-holomorphic curves of bidegree (4,16) on $F_4$ in a tame almost complex structure J, such that the exceptional J-holomorphic curve (i.e., the section E with E.E = - 4 ) exists.

[1]. S.Yu. Orevkov, E.I. Shustin. Flexible, algebraically unrealizable curves: rehabilitation of Hilbert-Rohn-Gudkov approach, J. fur die Reine und Angew. Math., 551(2002), 145-172