Corrigendum to “A flexible affine $M$-sextic which is algebraically unrealizable”
Authors:
S. Fiedler-Le Touzé, S. Orevkov and E. Shustin
Journal:
J. Algebraic Geom. 29 (2020), 109-121
DOI:
https://doi.org/10.1090/jag/733
Published electronically:
August 28, 2019
MathSciNet review:
4028067
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Abstract |
References |
Additional Information
Abstract: We prove the algebraic unrealizability of a certain isotopy type of plane affine real algebraic $M$-sextic which is pseudoholomorphically realizable. This result completes the classification up to isotopy of real algebraic affine $M$-sextics. The proof of this result given in a previous paper by the first two authors [J. Algebraic Geom. 11 (2002), pp. 293–310] was incorrect.
References
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- S. Fiedler-Le Touzé, Pencils of cubics with eight base points lying in convex position in $\mathbb {R}P^2$, arXiv [math AG] 1012.2679, 2010.
- S. Fiedler-Le Touzé, Rational pencils of cubics and configurations of six or seven points in $\mathbb {R}P^2$, arXiv[AG] 1210.7146, 2012.
- S. Fiedler-Le Touzé and S. Yu. Orevkov, A flexible affine $M$-sextic which is algebraically unrealizable, J. Algebraic Geom. 11 (2002), no. 2, 293–310. MR 1874116, DOI https://doi.org/10.1090/S1056-3911-2019-00733-2
- D. A. Gudkov, Variability of simple double points of real plane algebraic curves., Dokl. Akad. Nauk SSSR 142 (1962), 1233–1235 (Russian). MR 0146731
- D. A. Gudkov, Systems of $k$ points in general position and algebraic curves of different orders, Gor′kov. Gos. Univ. Učen. Zap. Vyp. 87 (1969), 21–58 (Russian). MR 0260736
- Ilia Itenberg, Viatcheslav Kharlamov, and Eugenii Shustin, Welschinger invariants revisited, Analysis meets geometry, Trends Math., Birkhäuser/Springer, Cham, 2017, pp. 239–260. MR 3773620
- A. B. Korchagin and E. I. Shustin, Sixth-degree affine curves and smoothings of a nondegenerate sixth-order singular point, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 1181–1199, 1327 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 3, 501–520. MR 984215, DOI https://doi.org/10.1070/IM1989v033n03ABEH000854
- S. Yu. Orevkov, Link theory and oval arrangements of real algebraic curves, Topology 38 (1999), no. 4, 779–810. MR 1679799, DOI https://doi.org/10.1016/S0040-9383%2898%2900021-4
- Stepan Yu. Orevkov, Riemann existence theorem and construction of real algebraic curves, Ann. Fac. Sci. Toulouse Math. (6) 12 (2003), no. 4, 517–531 (English, with English and French summaries). MR 2060598
- S. Yu. Orevkov, Positions of an $M$-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. MR 2381938, DOI https://doi.org/10.1090/S1061-0022-08-01014-5
- S. Yu. Orevkov and E. I. Shustin, Flexible, algebraically unrealizable curves: rehabilitation of Hilbert-Rohn-Gudkov approach, J. Reine Angew. Math. 551 (2002), 145–172. MR 1932177, DOI https://doi.org/10.1515/crll.2002.080
- S. Yu. Orevkov and E. I. Shustin, Pseudoholomorphic algebraically unrealizable curves, Mosc. Math. J. 3 (2003), no. 3, 1053–1083, 1200–1201 (English, with English and Russian summaries). {Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday}. MR 2078573, DOI https://doi.org/10.17323/1609-4514-2003-3-3-1053-1083
- S. Yu. Orevkov and E. I. Shustin, Real algebraic and pseudoholomorphic curves on the quadratic cone and smoothings of the singularity of $X_{21}$, Algebra i Analiz 28 (2016), no. 2, 138–186 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 28 (2017), no. 2, 225–257. MR 3593007, DOI https://doi.org/10.1090/S1061-0022-2017-01448-X
- E. I. Shustin, On manifolds of singular algebraic curves [translation of Methods of the qualitative theory of differential equations (Russian), 148–163, Gor′kov. Gos. Univ., Gorki, 1983; MR0834197 (87c:14026)], Selecta Math. Soviet. 10 (1991), no. 1, 27–37. Selected translations. MR 1099434
- O. Ya. Viro, Real plane algebraic curves: constructions with controlled topology, Algebra i Analiz 1 (1989), no. 5, 1–73 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 5, 1059–1134. MR 1036837
References
- Erwan Brugallé, Symmetric plane curves of degree 7: pseudoholomorphic and algebraic classifications, J. Reine Angew. Math. 612 (2007), 129–171. MR 2364076, DOI https://doi.org/10.1515/CRELLE.2007.086
- S. Fiedler-Le Touzé, Pencils of cubics with eight base points lying in convex position in $\mathbb {R}P^2$, arXiv [math AG] 1012.2679, 2010.
- S. Fiedler-Le Touzé, Rational pencils of cubics and configurations of six or seven points in $\mathbb {R}P^2$, arXiv[AG] 1210.7146, 2012.
- S. Fiedler-Le Touzé and S. Yu. Orevkov, A flexible affine $M$-sextic which is algebraically unrealizable, J. Algebraic Geom. 11 (2002), no. 2, 293–310. MR 1874116, DOI https://doi.org/10.1090/S1056-3911-01-00300-9
- D. A. Gudkov, Variability of simple double points of real plane algebraic curves (Russian), Dokl. Akad. Nauk SSSR 142 (1962), 1233–1235. MR 146731; English transl., Sov. Math., Dokl. 3 (1962), 273–275. Zbl 0119.37202
- D. A. Gudkov, Systems of $k$ points in general position and algebraic curves of different orders (Russian), Gor′kov. Gos. Univ. Učen. Zap. Vyp. 87 (1969), 21–58. MR 260736; English transl., Nine papers on Hilbert’s 16th problem, Amer. Math. Soc. Transl (2), vol. 112, 1978, pp. 15–45, DOI http://dx.doi.org/10.1090/trans2/112. MR 604783
- Ilia Itenberg, Viatcheslav Kharlamov, and Eugenii Shustin, Welschinger invariants revisited, Analysis meets geometry, Trends Math., Birkhäuser/Springer, Cham, 2017, pp. 239–260. MR 3773620
- A. B. Korchagin and E. I. Shustin, Sixth-degree affine curves and smoothings of a nondegenerate sixth-order singular point, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 1181–1199, 1327 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 3, 501–520. MR 984215, DOI https://doi.org/10.1070/IM1989v033n03ABEH000854
- S. Yu. Orevkov, Link theory and oval arrangements of real algebraic curves, Topology 38 (1999), no. 4, 779–810. MR 1679799, DOI https://doi.org/10.1016/S0040-9383%2898%2900021-4
- Stepan Yu. Orevkov, Riemann existence theorem and construction of real algebraic curves, Ann. Fac. Sci. Toulouse Math. (6) 12 (2003), no. 4, 517–531 (English, with English and French summaries). MR 2060598
- S. Yu. Orevkov, Positions of an $M$-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. MR 2381938, DOI https://doi.org/10.1090/S1061-0022-08-01014-5
- S. Yu. Orevkov and E. I. Shustin, Flexible, algebraically unrealizable curves: rehabilitation of Hilbert-Rohn-Gudkov approach, J. Reine Angew. Math. 551 (2002), 145–172. MR 1932177, DOI https://doi.org/10.1515/crll.2002.080
- S. Yu. Orevkov and E. I. Shustin, Pseudoholomorphic algebraically unrealizable curves, Mosc. Math. J. 3 (2003), no. 3, 1053–1083, 1200–1201 (English, with English and Russian summaries). MR 2078573
- S. Yu. Orevkov and E. I. Shustin, Real algebraic and pseudoholomorphic curves on the quadratic cone and smoothings of the singularity of $X_{21}$, Algebra i Analiz 28 (2016), no. 2, 138–186 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 28 (2017), no. 2, 225–257. MR 3593007, DOI https://doi.org/10.1090/spmj/1448
- E. I. Shustin, On manifolds of singular algebraic curves [translation of Methods of the qualitative theory of differential equations (Russian), 148–163, Gor ′kov. Gos. Univ., Gorki, 1983; MR0834197 (87c:14026)], Selected translations, Selecta Math. Soviet. 10 (1991), no. 1, 27–37. MR 1099434
- O. Ya. Viro, Real plane algebraic curves: constructions with controlled topology, Algebra i Analiz 1 (1989), no. 5, 1–73 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 5, 1059–1134. MR 1036837
Additional Information
S. Fiedler-Le Touzé
Affiliation:
IMT, l’université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
Email:
severine.fiedler@live.fr
S. Orevkov
Affiliation:
IMT, l’université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France; and Steklov Mathematics Institut, Gubkina 8, 119991 Moscow, Russia
MR Author ID:
202757
Email:
orevkov@math.ups-tlse.fr
E. Shustin
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel
MR Author ID:
193452
Email:
shustin@math.tau.ac.il
Received by editor(s):
January 14, 2018
Received by editor(s) in revised form:
June 26, 2018
Published electronically:
August 28, 2019
Additional Notes:
The second author was partially supported by RFBR grant no. 17-01-00592a. The third author was supported by the Israeli Science Foundation grant no. 176/15.
Article copyright:
© Copyright 2019
University Press, Inc.
