Geodesic
On the topology of planar real decomposable curves of degree~8
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the XVII All-Russian Youth School-Conference «Lobachevsky Readings-2018», November 23-28, 2018, Kazan. Part 2, Tome 176 (2020), pp. 3-18.

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We consider the problem of topological classification of arrangements in the real projective plane of the union of nonsingular curves of degrees $2$ and $6$ under certain conditions of maximality and general position. We list admissible topological models of such arrangements to be studied by using the Orevkov method based on the theory of braids and links and prove that most of these models cannot be realized by curves of degree $8$.
Mots-clés : $M$-decomposable curve
Keywords: topological classification, Orevkov method. 
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I. M. Borisov; G. M. Polotovsky. On the topology of planar real decomposable curves of degree~8. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the XVII All-Russian Youth School-Conference «Lobachevsky Readings-2018», November 23-28, 2018, Kazan. Part 2, Tome 176 (2020), pp. 3-18. http://geodesic.mathdoc.fr/item/INTO_2020_176_a0/

[31] Viro O. Ya., “Ploskie veschestvennye algebraicheskie krivye: postroeniya s kontroliruemoi topologiei”, Algebra i analiz., 1:5 (1989), 1–73

[32] Gudkov D. A., “Topologiya veschestvennykh proektivnykh algebraicheskikh mnogoobrazii”, Usp. mat. nauk., 29:4 (178) (1974), 3–79 | MR | Zbl

[33] Gudkov D. A., Utkin G. A., “Topologiya krivykh 6-go poryadka i poverkhnostei 4-go poryadka (k 16-i probleme Gilberta)”, Uch. zap. Gorkov. un-ta., 87 (1969), 1–214

[34] Guschin M. A., “Postroeniya nekotorykh raspolozhenii koniki i $M$-kvintiki s odnoi tochkoi na beskonechnosti”, Vestn. NNGU. Ser. mat., 2004, no. 1 (2), 43–52

[35] Guschin M. A., “Konika i $M$-kvintika s odnoi tochkoi na beskonechnosti”, Zap. nauch. semin. POMI., 329 (2005), 14–27

[36] Guschin M. A., Korobeinikov A. N., Polotovskii G. M., “Postroenie vzaimnykh raspolozhenii kubiki i kvartiki metodom kusochnogo konstruirovaniya”, Zap. nauch. semin. POMI., 267 (2000), 119–132

[37] Korobeinikov A. N., “Novye postroeniya raspadayuschikhsya krivykh”, Vestn. NNGU. Ser. mat. model. optim. upravl., 2001, no. 1(23), 17–27

[38] Korchagin A. B., Polotovskii G. M., “O raspolozheniyakh ploskoi veschestvennoi kvintiki otnositelno pary pryamykh”, Algebra i analiz., 21:2 (2009), 92–112 | MR

[39] Korchagin A. B., Shustin E. I., “Affinnye krivye stepeni 6 i ustraneniya nevyrozhdennoi shestikratnoi osoboi tochki”, Izv. AN SSSR. Ser. mat., 52:6 (1988), 1181–1199

[40] Orevkov S. Yu., “Novaya affinnaya $M$-sekstika”, Funkts. anal. prilozh., 32 (1998), 141–143 | Zbl

[41] Orevkov S. Yu., “Novaya affinnaya $M$-sekstika, II”, Usp. mat. nauk., 53:5(323) (1998), 243–244 | DOI | MR | Zbl

[42] Orevkov S. Yu., “Proektivnye koniki i $M$-kvintiki v obschem polozhenii s maksimalno peresekayuscheisya paroi ovalov”, Mat. zametki., 65:4 (1999), 632–635 | DOI

[43] Orevkov S. Yu., “Postroenie raspolozhenii $M$-kvartiki i $M$-kubiki s maksimalno peresekayuschimisya ovalom i nechetnoi vetvyu”, Vestn. NNGU. Ser. mat. model. optim. upravl., 2002, no. 1 (25), 12–48

[44] Orevkov S. Yu., “Raspolozheniya $M$-kvintiki otnositelno koniki, maksimalno peresekayuschei ee nechetnuyu vetv”, Algebra i analiz., 19:4 (2007), 174–242

[45] Orevkov S. Yu., Polotovskii G. M., “Proektivnye $M$-kubiki i $M$-kvartiki v obschem polozhenii s maksimalno peresekayuscheisya paroi ovalov”, Algebra i analiz., 11:5 (1999), 166–184

[46] Polotovskii G. M., “K zadache topologicheskoi klassifikatsii raspolozheniya ovalov neosobykh algebraicheskikh krivykh v proektivnoi ploskosti”, Metody kachestvennoi teorii differentsialnykh uravnenii, Izd-vo GGU, Gorkii, 1975, 101–128

[47] Polotovskii G. M., “Katalog $M$-raspadayuschikhsya krivykh 6-go poryadka”, Dokl. AN SSSR., 236:3 (1977), 548–551 | MR | Zbl

[48] Polotovskii G. M., Polnaya klassifikatsiya $M$-raspadayuschikhsya krivykh $6$-go poryadka v veschestvennoi proektivnoi ploskosti, Dep. v VINITI. No 1349-78, M., 1978

[49] Polotovskii G. M., $(M - 2)$-Krivye 8-go poryadka: postroeniya, otkrytye voprosy, Dep. v VINITI. No 1185-85, M., 1985

[50] Shustin E. I., “K izotopicheskoi klassifikatsii affinnykh$M$-krivykh stepeni $6$”, Metody kachestvennoi teorii i teorii bifurkatsii, Izd-vo GGU, Gorkii, 1988, 97–105

[51] Binstein A. A., Polotovskii G. M., “On the mutual arrangement of a conic and a quintic in the real projective plane”, Methods of Qualitative Theory of Differential Equations and Related Topics. Am. Math. Ser. Transl. Ser. 2, v. 200, 2000, 63–72 | MR | Zbl

[52] Fiedler-Le Touz'e S., Orevkov S. Yu., “Flexible affine $M$-sextic which is algebraically unrealizable”, J. Alg. Geom., 11 (2002), 293–310 | DOI | MR | Zbl

[53] Fiedler-Le Touz'e S., Orevkov S. Yu., Shustin E. I., Corrigendum to the paper “A flexible affine M-sextic which is algebraically unrealizable”, arXiv: 1801.04905 [math.AG] | MR

[54] Korchagin A. B., Polotovskii G. M., “On arrangements of a plane real quintic curve with respect to a pair of lines”, Commun. Contemp. Math., 5:1 (2003), 1–24 | DOI | MR | Zbl

[55] Lee R., “Algebraic functions and closed braids”, Topology., 22 (1983), 191–202 | DOI | MR | Zbl

[56] Orevkov S. Yu., “Link theory and oval arrangements of real algebraic curve”, Topology., 38 (1999), 779–810 | DOI | MR | Zbl

[57] Orevkov S. Yu., “Clasification flexible $M$-curves of degree 8 up to isotopy”, Geom. Funct. Anal., 12:4 (2002), 723–755 | DOI | MR | Zbl

[58] Orevkov S. Yu., Shustin E. I., “Flexible, algebraically unrealizable curves: rehabilitation of the Hilbert–Rohn–Gudkov approach”, J. Reine Angew. Math., 551 (2002), 145–172 | MR | Zbl

[59] Orevkov S. Yu., Shustin E. I., “Pseudoholomorphic algebraically unrealizable curves”, Moscow Math. J., 3:3 (2003), 1053–1083 | DOI | MR | Zbl

[60] Polotovskii G. M., “On the classification of decomposing plane algebraic curves”, Lect. Notes Math., 1524 (1992), 52–74 | DOI | MR

[61] Polotovskii G. M., “On the classification of decomposable 7-th degree curves”, Contemp. Math., 253 (2000), 219–234 | DOI | MR | Zbl

[62] Polotovskiy G. M., “On the classification of 7th degree real decomposable curves”, Adv. Stud. Pure Math., 43 (2006), 369–382 | DOI | MR | Zbl