Journal of Algebraic Geometry

Author(s): S. Fiedler-Le Touzé; S. Yu. Orevkov
Journal: J. Algebraic Geom. 11 (2002), 293-310.
Posted: December 13, 2001
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Abstract: 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 pseudo-holomorphic in a suitable tame almost complex structure on $\mathbf{CP}^{2}$ .

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.

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S. Fiedler-Le Touzé
Affiliation: Laboratoire E. Picard, UFR MIG, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France
Email: fiedler@picard.ups-tlse.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
Email: orevkov@picard.ups-tlse.fr

PII: S 1056-3911(01)00300-9
Received by editor(s): December 15, 1999
Received by editor(s) in revised form: July 4, 2000
Posted: December 13, 2001

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