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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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.
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.
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
MR Author ID:
202757
Email:
orevkov@picard.ups-tlse.fr
Received by editor(s):
December 15, 1999
Received by editor(s) in revised form:
July 4, 2000
Published electronically:
December 13, 2001
