A flexible affine M-sextic which is algebraically unrealizable

by S. Fiedler-LeTouzé and S.Yu. Orevkov

We prove that a cetrain arrangement of a line on $RP^2$ and 11 embedded circles (ovals) can not be isotopic to the union of a real line a real algebraic curve of degree six. Since the line can be considered as the line at infinity, this is a restriction for the arrangement of a real affine sextic on the real affine plane.

Previously, the second author realized this arrangement with flexible curves. Here we show that these flexible curves are pseudo-holomorphic in a suitable tame almost complex structure on $CP^2$.

For the proof of the algebraic non-realizability we consider all possible positions of the curve with respect to certain pencils of lines. Using the Murasugi-Tristram inequality for certain links in $S^3$, we show that all the positions but one are unrealizable. Then, we prohibit the last position (the one which is realizable by a flexible curve) by studying its behaviour with respect to an auxiliary pencil of cubics.