Real algebraic curves of bidegree $(5,5)$ on the quadric ellipsoid

M. Manzaroli

Algebra i analiz, Tome 32 (2020) no. 2, pp. 107-142.

Résumé

In this paper, the topological classification of nonseparating (respectively, separating) real algebraic nonsingular $(M-i)$-curves of bidegree $ (5, 5)$ on the quadric ellipsoid is completed. In particular, it is shown that previously known restrictions form a complete system for this bidegree. Therefore, the main part of the paper concerns the construction of real algebraic curves. The strategy is to reduce the problem of construction of curves on the quadric ellipsoid to construction of curves on the second Hirzebruch surface by degenerating the quadric ellipsoid to the quadratic cone. Next, various classical construction methods on toric surfaces are combined, such as dessins d'enfants and Viro's patchworking method.

Détail

Keywords: real algebraic variety, Hilbert's 16th problem, topology of real algebraic curves, quadric ellipsoid.

Bibliographie

[1] Brugallé E., “Symmetric plane curves of degree $7$: pseudoholomorphic and algebraic classifications”, J. Reine Angew. Math., 612 (2007), 129–171 | MR | Zbl

[2] Degtyarev A. I., Itenberg I., Kharlamov V. M., “On deformation types of real elliptic surfaces”, Amer. J. Math., 6:6 (2008), 1561–1627 | DOI | MR | Zbl

[3] Degtyarev A. I., Itenberg I., Zvonilov V. I., “Real trigonal curves and real elliptic surfaces of type I”, J. Reine Angew. Math., 686 (2014), 221–246 | MR | Zbl

[4] Degtyarev A.I., Kharlamov V.M., “Topologicheskie svoistva veschestvennykh algebraicheskikh mnogoobrazii: du cot̂è de chez Rokhlin”, Uspekhi mat. nauk, 55:4 (2000), 129–212 | DOI | MR | Zbl

[5] Degtyarev A. I., Zvonilov V. I., “Zhestkaya izotopicheskaya klassifikatsiya veschestvennykh algebraicheskikh krivykh bistepeni $(3,3)$ na kvadrikakh”, Mat. zametki, 66:6 (1999), 810–815 | DOI | Zbl

[6] Guillou L., Marin A., “Une extension d'un théorème de Rohlin sur la signature”, C. R. Acad. Sci. Paris Sér. A-B, 285:3 (1977), A95–A98 | MR

[7] Gudkov D. A., Shustin E. I., “Klassifikatsiya neosobykh krivykh vosmogo poryadka na ellipsoide”, Metody kachestvennoi teorii differentsialnykh uravnenii, Gork. gos. un-t, Gorkii, 1980, 104–107

[8] Harnack A., “Über vieltheiligkeit der ebenen algebraischen curven”, Math. Ann., 10:2 (1876), 189–198 | DOI | MR

[9] Jaramillo Puentes A., Rigid isotopy classification of generic rational quintics in $\mathbb{R}P^2$, 2018, arXiv: 1804.04982 [math.AG]

[10] Klein F., Gesammelte mathematische Abhandlungen, v. 2, Springer, Berlin, 1922 | MR

[11] Mikhalkin G., “Congruences for real algebraic curves on an ellipsoid”, Topology of manifolds and varieties, Adv. Soviet Math., 18, Amer. Math. Soc., Providence, RI, 1994, 223–233 | MR | Zbl

[12] Nikulin V. V., “Filtratsii $2$-elementarnykh form i involyutsii tselochislennykh bilineinykh simmetricheskikh i kososimmetricheskikh form”, Izv. AN SSSR. Ser. mat., 49:4 (1985), 847–873 | MR | Zbl

[13] Nikulin V. V., Saito S., “Real $K3$ surfaces with non-symplectic involution and applications”, Proc. London Math. Soc. (3), 90:3 (2005), 591–654 | DOI | MR | Zbl

[14] Nikulin V. V., Saito S., “Real $K3$ surfaces with non-symplectic involution and applications II”, Proc. Lond. Math. Soc. (3), 95:1 (2007), 20–48 | DOI | MR | Zbl

[15] Orevkov S. Y., “Riemann existence theorem and construction of real algebraic curves”, Ann. Fac. Sci. Toulouse Math. (6), 12:4 (2003), 517–531 | DOI | MR | Zbl

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

[17] Orevkov S. Yu., Shustin E. I., “Veschestvennye algebraicheskie i psevdogolomorfnye krivye na kvadratichnom konuse i sglazhivaniya osobennosti $X_{21}$”, Algebra i analiz, 28:2 (2016), 138–186

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

[19] Rokhlin V. A., “Sravneniya po modulyu $16$ v shestnadtsatoi probleme Gilberta”, Funkts. anal. i ego pril., 6:4 (1972), 58–64 | MR

[20] Shustin E. I., “Metod Gilberta–Roona i bifurkatsii slozhnykh osobykh tochek krivykh vosmogo poryadka”, Uspekhi mat. nauk, 38:6 (1983), 157–158

[21] Shustin E. I., “Gluing of singular and critical points”, Topology, 37:1 (1998), 195–217 | DOI | MR | Zbl

[22] Shustin E. I., Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry, , 2002 math/0211278 [math.AG]

[23] Shustin E., “A tropical approach to enumerative geometry”, Algebra i analiz, 17:2 (2005), 170–214 | MR

[24] Shustin E. I., “The patchworking construction in tropical enumerative geometry”, Singularities and computer algebra, London Math. Soc. Lecture Note Ser., 324, Cambridge Univ. Press, Cambridge, 2006, 273–300 | MR | Zbl

[25] Viro O. Y., “Gluing of plane real algebraic curves and constructions of curves of degrees $6$ and $7$”, Topology (Leningrad, 1982), Lecture Notes in Math., 1060, Springer, Berlin, 1984, 187–200 | DOI | MR

[26] Viro O. Ya., “Ploskie veschestvennye krivye stepenei $7$ i $8$: novye zaprety”, Izv. AN SSSR. Ser. mat., 47:5 (1983), 1135–115 | MR

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

[28] Viro O. Y., Patchworking real algebraic varieties, 2006, arXiv: math/0611382 [math.AG]

[29] Zvonilov V. I., “Kompleksnye orientatsii veschestvennykh algebraicheskikh krivykh s osobennostyami”, Dokl. AN SSSR, 268:1 (1983), 22–26 | MR | Zbl

[30] Zvonilov V.I., “Kompleksnye topologicheskie invarianty veschestvennykh algebraicheskikh krivykh na giperboloide i ellipsoide”, Algebra i analiz, 3:3 (1991), 88–108 | MR | Zbl