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 Mosc. Math. J., 2003, Volume 3, Number 3, Pages 1053–1083 (Mi mmj121)

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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Bibliographic databases:

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
\Bibitem{OreShu03} \by S.~Yu.~Orevkov, E.~I.~Shustin \paper Pseudoholomorphic algebraically unrealizable curves \jour Mosc. Math.~J. \yr 2003 \vol 3 \issue 3 \pages 1053--1083 \mathnet{http://mi.mathnet.ru/mmj121} \crossref{https://doi.org/10.17323/1609-4514-2003-3-3-1053-1083} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2078573} \zmath{https://zbmath.org/?q=an:1049.14044} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208594300014} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. 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
2. 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
3. Brugalle E., “Symmetric plane curves of degree 7: pseudoholomorphic and algebraic classifications”, Journal fur Die Reine und Angewandte Mathematik, 612 (2007), 129–171
4. Bertrand B., Brugalle E., “A nonalgebraic patchwork”, Mathematische Zeitschrift, 259:3 (2008), 481–486
5. 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
6. 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