Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

A flexible affine $M$-sextic which is algebraically unrealizable


Authors: S. Fiedler-Le Touzé and S. Yu. Orevkov
Journal: J. Algebraic Geom. 11 (2002), 293-310
DOI: https://doi.org/10.1090/S1056-3911-01-00300-9
Published electronically: December 13, 2001
MathSciNet review: 1874116
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Abstract | References | Additional Information

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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Additional Information

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

DOI: https://doi.org/10.1090/S1056-3911-01-00300-9
Received by editor(s): December 15, 1999
Received by editor(s) in revised form: July 4, 2000
Published electronically: December 13, 2001

American Mathematical Society