On arrangements of plane real quintics with respect to a pair of lines.

*(English. Russian original)*Zbl 1206.14057
St. Petersbg. Math. J. 21, No. 2, 231-244 (2010); translation from Algebra Anal. 21, No. 2, 92-112 (2009).

A real projective curve of degree \(n\) is a real homogeneous polynomial \(C_n\in\mathbb R[X,Y,Z]\) defined up to a nonzero constant divisor. Its set of real zeroes in the real projective plane \(\mathbb R{ \mathbb{P}}^2\) is denoted by \(\mathbb R C_n\).

Let \(C_1\) and \(C'_1\) be two distinct lines. Hence, the set \(\mathbb R{ \mathbb{P}}^2\setminus(\mathbb R C_1\cup\mathbb R C'_1))\) consists of two connected components, each homeomorphic to an open disk, and whose closures are denoted by \(D_1\) and \(D_2\). In this paper the authors complete the topological classification of the quadruples \[ (\mathbb R{ \mathbb{P}}^2,\mathbb R C_5\cup\mathbb R C_1 \cup \mathbb R C'_1, \mathbb R C_5\cup\mathbb R C_1,\mathbb R C_1) \] under the assumptions of maximality and general position. This means the following:

(1) The quintic curve \(C_5\) is an \(M\)-curve, that is, \(\mathbb R C_5\) consists of an odd branch \(J\) and \(6\) ovals that do not surround one another.

(2) \(\mathbb R C_5,\mathbb R C_1\) and \(\mathbb R C'_1\) are in general position, that is, they have pairwise transversal intersection and \(\mathbb R C_5\) does not pass through \(\mathbb R C_1 \cap \mathbb R C'_1\).

(3) The intersections \(\mathbb R C_1\cap J\) and \(\mathbb R C'_1\cap J\) have \(5\) distinct points.

This article completes the topological classification of quadruples in (i). Let us explain this more in detail. Let \(a\) and \(b\) be the number of arcs of \(J\) lying on \(D_1\) and \(D_2\), respectively, and having one end point lying on \(\mathbb R C_1\) and the other in \(\mathbb R C'_1\). The classification problem was solved by the authors in case \(a=b=1\) in their previous work [Commun. Contemp. Math. 5, No. 1, 1–24 (2003; Zbl 1077.14568)]. Moreover, the solution if \(b=5\) and either \(a=1\) or \(a=3\) follows at once from the results in [G. M. Polotovskii, Sov. Math., Dokl. 18(1977), 1241–1245 (1978); translation from Dokl. Akad. Nauk SSSR 236, 548–551 (1977; Zbl 0392.14014)]. Moreover, the classification was obtained by S. Yu. Orevkov [St. Petersbg. Math. J. 19, No. 4, 625–674 (2008); translation from Algebra Anal. 19, No. 4, 174–242 (2007; Zbl 1206.14082)]. Thus the only case that remained unsolved was \((a,b)=(1,3)\), and this is the goal of the paper under review.

The answer to the classification problem is rather complicated to be reproduced here. I can say that there exist \(51\) types of topological models for (i) above and that the proof is based on a similar approach to one used by the authors in its joint paper quoted above. In particular, the constructions needed to realize the existence of some configurations use Viro’s patchworking method.

The authors recognize explicitly in the paper under review the contribution of Orevkov to the final form of the classification.

Let \(C_1\) and \(C'_1\) be two distinct lines. Hence, the set \(\mathbb R{ \mathbb{P}}^2\setminus(\mathbb R C_1\cup\mathbb R C'_1))\) consists of two connected components, each homeomorphic to an open disk, and whose closures are denoted by \(D_1\) and \(D_2\). In this paper the authors complete the topological classification of the quadruples \[ (\mathbb R{ \mathbb{P}}^2,\mathbb R C_5\cup\mathbb R C_1 \cup \mathbb R C'_1, \mathbb R C_5\cup\mathbb R C_1,\mathbb R C_1) \] under the assumptions of maximality and general position. This means the following:

(1) The quintic curve \(C_5\) is an \(M\)-curve, that is, \(\mathbb R C_5\) consists of an odd branch \(J\) and \(6\) ovals that do not surround one another.

(2) \(\mathbb R C_5,\mathbb R C_1\) and \(\mathbb R C'_1\) are in general position, that is, they have pairwise transversal intersection and \(\mathbb R C_5\) does not pass through \(\mathbb R C_1 \cap \mathbb R C'_1\).

(3) The intersections \(\mathbb R C_1\cap J\) and \(\mathbb R C'_1\cap J\) have \(5\) distinct points.

This article completes the topological classification of quadruples in (i). Let us explain this more in detail. Let \(a\) and \(b\) be the number of arcs of \(J\) lying on \(D_1\) and \(D_2\), respectively, and having one end point lying on \(\mathbb R C_1\) and the other in \(\mathbb R C'_1\). The classification problem was solved by the authors in case \(a=b=1\) in their previous work [Commun. Contemp. Math. 5, No. 1, 1–24 (2003; Zbl 1077.14568)]. Moreover, the solution if \(b=5\) and either \(a=1\) or \(a=3\) follows at once from the results in [G. M. Polotovskii, Sov. Math., Dokl. 18(1977), 1241–1245 (1978); translation from Dokl. Akad. Nauk SSSR 236, 548–551 (1977; Zbl 0392.14014)]. Moreover, the classification was obtained by S. Yu. Orevkov [St. Petersbg. Math. J. 19, No. 4, 625–674 (2008); translation from Algebra Anal. 19, No. 4, 174–242 (2007; Zbl 1206.14082)]. Thus the only case that remained unsolved was \((a,b)=(1,3)\), and this is the goal of the paper under review.

The answer to the classification problem is rather complicated to be reproduced here. I can say that there exist \(51\) types of topological models for (i) above and that the proof is based on a similar approach to one used by the authors in its joint paper quoted above. In particular, the constructions needed to realize the existence of some configurations use Viro’s patchworking method.

The authors recognize explicitly in the paper under review the contribution of Orevkov to the final form of the classification.

Reviewer: Jose Manuel Gamboa (Madrid)

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\textit{A. B. Korchagin} and \textit{G. M. Polotovskiĭ}, St. Petersbg. Math. J. 21, No. 2, 231--244 (2010; Zbl 1206.14057); translation from Algebra Anal. 21, No. 2, 92--112 (2009)

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##### References:

[1] | Anatoly B. Korchagin and Grigory M. Polotovskii, On arrangements of a plane real quintic curve with respect to a pair of lines, Commun. Contemp. Math. 5 (2003), no. 1, 1 – 24. · Zbl 1077.14568 |

[2] | G. M. Polotovskiĭ, A catalogue of \?-splitting curves of order six, Dokl. Akad. Nauk SSSR 236 (1977), no. 3, 548 – 551 (Russian). |

[3] | S. Yu. Orevkov, Positions of an \?-quintic with respect to a conic that maximally intersect the odd branch of the quintic, Algebra i Analiz 19 (2007), no. 4, 174 – 242 (Russian); English transl., St. Petersburg Math. J. 19 (2008), no. 4, 625 – 674. |

[4] | S. Yu. Orevkov, Link theory and oval arrangements of real algebraic curves, Topology 38 (1999), no. 4, 779 – 810. · Zbl 0923.14032 |

[5] | O. Ya. Viro, Gluing of algebraic hypersurfaces, elimination of singularities, and constructions of curves, Proceedings of the International Topological Conference (Leningrad, 1982), Nauka, Leningrad, 1983, pp. 149-197. (Russian) |

[6] | O. Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Topology (Leningrad, 1982) Lecture Notes in Math., vol. 1060, Springer, Berlin, 1984, pp. 187 – 200. · Zbl 0576.14031 |

[7] | -, Real algebraic manifolds with prescribed topological properties, Doctor’s Diss., Leningrad. Gos. Univ., Leningrad, 1983. (Russian) |

[8] | A. B. Korchagin, Smoothing of 6-fold singular points and constructions of 9th degree \?-curves, Topology of real algebraic varieties and related topics, Amer. Math. Soc. Transl. Ser. 2, vol. 173, Amer. Math. Soc., Providence, RI, 1996, pp. 141 – 155. · Zbl 0858.14029 |

[9] | A. B. Korchagin and D. E. Smith, Patchworking singularities \?_{\?} and \?_{\?} and meanders of their smoothing, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 299 (2003), no. Geom. i Topol. 8, 193 – 217, 330 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 131 (2005), no. 1, 5366 – 5380. · Zbl 1140.14321 |

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