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.


References [Enhancements On Off] (What's this?)

  • 1. S. Fiedler-Le Touzé, Orientations complexes des courbes algébriques réelles, Thèse doctorale (1999).
  • 2. C.McA. Gordon, R.A. Litherland, On the signature of a link, Invent. Math. 47 (1978), 53-69.
  • 3. M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347.
  • 4. A.B. Korchagin, E.I. Shustin, Affine curves of degree 6 and smoothing of non-degenerate six-fold singular points, Math. USSR-Izvestia 33 (1989), 501-520.
  • 5. S.Yu. Orevkov, Link theory and oval arrangements of real algebraic curves, Topology 38 (1999), 779-810.
  • 6. S.Yu. Orevkov, A new affine M-sextic, Russ. Math. Surv. 53 (1999), 1099-1101,
  • 7. G.M. Polotovskii, $(M-2)$-curves of 8-th order: constructions, open questions, Deponent VINITI, N1185-85, 1984, 1-194.
  • 8. G. Ringel, Teilungen der Ebene durch Geraden oder topologische Geraden, Math. Z. 64 (1956), 79-102.
  • 9. Rokhlin V.A., Complex topological characteristics of real algebraic curves, Russ. Math. Surv. 33:5 (1978), 85-98.
  • 10. E.I. Shustin, New $M$-curve of 8th degree, Math. Notes 42 (1987), 606-610.
  • 11. O.Ya. Viro, Progress in the topology of real algebraic varieties over the last six years, Russian Math. Surveys 41 (1986), 55-82.


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

Journal of Algebraic Geometry
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