A flexible affine $M$sextic which is algebraically unrealizable
Authors:
S. FiedlerLe Touzé and S. Yu. Orevkov
Journal:
J. Algebraic Geom. 11 (2002), 293310
DOI:
https://doi.org/10.1090/S1056391101003009
Published electronically:
December 13, 2001
MathSciNet review:
1874116
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Abstract  References  Additional Information
We prove that the union of a real algebraic curve of degree six and a real line on $\mathbf {RP}^{2}$ 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 pseudoholomorphic in a suitable tame almost complex structure on $\mathbf {CP}^{2}$.
For the proof of the algebraic nonrealizability we consider all possible positions of the curve with respect to certain pencils of lines. Using the MurasugiTristram 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.

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Additional Information
S. FiedlerLe Touzé
Affiliation:
Laboratoire E. Picard, UFR MIG, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France
Email:
fiedler@picard.upstlse.fr
S. Yu. Orevkov
Affiliation:
Laboratoire E. Picard, UFR MIG, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France;
Steklov Institute of Mathematics, Vavilova 42, 117966 Moscow GSP/1, Russia
MR Author ID:
202757
Email:
orevkov@picard.upstlse.fr
Received by editor(s):
December 15, 1999
Received by editor(s) in revised form:
July 4, 2000
Published electronically:
December 13, 2001