We prove that the union of a real algebraic curve of degree six and a real line on RP 2 cannot be isotopic to the arrangement in Figure 1. 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.
We prove the algebraic unrealizability of certain isotopy type of plane affine real algebraic M-sextic which is pseudoholomorphically realizable. This result completes the classification up to isotopy of real algebraic affine M-sextics. The proof of this result given in a previous paper by the first two authors was incorrect.
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