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Real algebraic and pseudoholomorphic curves on the quadratic cone and smoothings of the singularity $ X_{21}$


Authors: S. Yu. Orevkov and E. I. Shustin
Translated by: the authors
Original publication: Algebra i Analiz, tom 28 (2016), nomer 2.
Journal: St. Petersburg Math. J. 28 (2017), 225-257
MSC (2010): Primary 14P25, 57M25; Secondary 14H20, 53D99
DOI: https://doi.org/10.1090/spmj/1448
Published electronically: February 15, 2017
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Abstract: A fiberwise isotopy classification is completed for the smooth real algebraic and pseudoholomorphic curves of degree 8 on the quadratic cone that have a specially shaped oval crossing a given generating line of the cone at four real points. This classification is linked with an isotopy classification of smoothings of a real plane curve singularity that is the union of four smooth real local branches quadratically tangent to each other (the singularity $ X_{21}$).


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Additional Information

S. Yu. Orevkov
Affiliation: Steklov Mathematical Institute, Gubkina 8, 119991 Moscow, Russia; IMT, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France; National Research University Higher School of Economics, Vavilova 7, 117312 Moscow, Russia
Email: orevkov@math.ups-tlse.fr

E. I. Shustin
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel
Email: shustin@math.tau.ac.il

DOI: https://doi.org/10.1090/spmj/1448
Received by editor(s): September 1, 2015
Published electronically: February 15, 2017
Additional Notes: The first author has been supported by RSF grant, project 14-21-00053 dated 11.08.14
The second author has been supported by the German–Israeli Foundation, grant no. 1174-197.6/2011, and by the Hermann–Minkowski–Minerva Center for Geometry at the Tel Aviv University. The main part of this work was performed during the second author’s visit to the Centre Interfacultaire Bernoulli at the École Polytechnique Fedérale de Lausanne in March-May 2015 and to the Max-Planck Institut für Mathematik, Bonn, in August-September 2015. The second author is very grateful to CIB-EPFL and MPI for hospitality and excellent working conditions. Special thanks are due to the referee, who pointed out several mistakes in the preliminary version of the paper
Dedicated: Dedicated to Sergeĭ Vladimirovich Vostokov, the first supervisor of the second author
Article copyright: © Copyright 2017 American Mathematical Society