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

Full-text PDF

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.

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