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This article is cited in 7 scientific papers (total in 7 papers)
Pseudoholomorphic algebraically unrealizable curves
S. Yu. Orevkova, E. I. Shustinb
a Université Paul Sabatier
b Tel Aviv University, School of Mathematical Sciences
Abstract:
We show that there exists a real non-singular pseudoholomorphic sextic curve in the affine plane which is not isotopic to any real algebraic sextic curve. This result completes the isotopy classification of real algebraic affine $M$-curves of degree 6. Comparing this with the isotopy classification of real affine pseudoholomorphic sextic $M$-curves obtained earlier by the first author, we find three pseudoholomorphic isotopy types which are algebraically unrealizable. In a similar way, we find a real pseudoholomorphic, algebraically unrealizable $(M-1)$-curve of degree 8 on a quadratic cone arranged in a special way with respect to a generating line. The proofs are based on the Hilbert–Rohn–Gudkov approach developed by the second author and on the cubic resolvent method developed by the first author.
Key words and phrases:
Pseudoholomorphic curves, real algebraic curves, equisingular family, cubic resolvent.
DOI:
https://doi.org/10.17323/1609-4514-2003-3-3-1053-1083
Full text:
http://www.ams.org/.../abst3-3-2003.html
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MSC: Primary 14P25, 57M25; Secondary 14H20, 53D99
Received: July 1, 2002; in revised form May 7, 2003
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Citation:
S. Yu. Orevkov, E. I. Shustin, “Pseudoholomorphic algebraically unrealizable curves”, Mosc. Math. J., 3:3 (2003), 1053–1083
Citation in format AMSBIB
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\paper Pseudoholomorphic algebraically unrealizable curves
\jour Mosc. Math.~J.
\yr 2003
\vol 3
\issue 3
\pages 1053--1083
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http://mi.mathnet.ru/eng/mmj121 http://mi.mathnet.ru/eng/mmj/v3/i3/p1053
Citing articles on Google Scholar:
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This publication is cited in the following articles:
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Lehavi D., “Mikhalkin's classification of M-curves in maximal position with respect to three lines”, Snowbird Lectures in Algebraic Geometry, Contemporary Mathematics Series, 388, 2005, 107–118
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S. Yu. Orevkov, “Arrangements of an $M$-quintic with respect to a conic that maximally intersects its odd branch”, St. Petersburg Math. J., 19:4 (2008), 625–674
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Brugalle E., “Symmetric plane curves of degree 7: pseudoholomorphic and algebraic classifications”, Journal fur Die Reine und Angewandte Mathematik, 612 (2007), 129–171
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Bertrand B., Brugalle E., “A nonalgebraic patchwork”, Mathematische Zeitschrift, 259:3 (2008), 481–486
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Shustin E., “Tropical and Algebraic Curves with Multiple Points”, Perspectives in Analysis, Geometry, and Topology: on the Occasion of the 60th Birthday of Oleg Viro, Progress in Mathematics, 296, eds. Itenberg I., Joricke B., Passare M., Birkhauser Verlag Ag, 2012, 431–464
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S. Yu. Orevkov, E. I. Shustin, “Real algebraic and pseudoholomorphic curves on the quadratic cone and smoothings of singularity $X_{21}$”, St. Petersburg Math. J., 28:2 (2017), 225–257
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I. M. Borisov, G. M. Polotovskii, “O topologii ploskikh veschestvennykh raspadayuschikhsya krivykh stepeni 8”, Materialy XVII Vserossiiskoi molodezhnoi shkoly-konferentsii «Lobachevskie chteniya-2018»,
23-28 noyabrya 2018 g., Kazan. Chast 2, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 176, VINITI RAN, M., 2020, 3–18
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