MOSCOW MATHEMATICAL JOURNAL Volume 3, Number 3, July–September 2003, Pages 1053–1083 PSEUDOHOLOMORPHIC ALGEBRAICALLY UNREALIZABLE CURVES S. YU. OREVKOV AND E. I. SHUSTIN To Vladimir Igorevich Arnold with admiration Abstract. We show that there exists a real non-singular pseudoholo- morphic 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. 2000 Math. Subj. Class. Primary 14P25, 57M25; Secondary 14H20, 53D99. Key words and phrases. Pseudoholomorphic curves, real algebraic curves, equisingular family, cubic resolvent. Introduction This paper can be considered as a continuation of [15]. The study of plane real pseudoholomorphic curves has been initiated by the first author [3], [11], [12]. It gives a new insight into the classical Hilbert 16th problem, whose first part is the question of classification of the oval arrangements of real plane non-singular alge- braic curves (for basic results and ideas see [1], [2], [7], [11], [17], [18], [24], [25], [27]). This classification is completed only for small degrees. The global rigidity of the symplectic structure, which becomes especially apparent in Gromov’s theory of pseudoholomorphic curves [5], makes the isotopy classifications 1 of real pseudo- holomorphic and algebraic curves rather similar. On the other hand, the difference between pseudoholomorphic and algebraic curves and methods for distinguishing Received July 1, 2002; in revised form May 7, 2003. Supported by the French-Israeli scientific cooperation program “Arc-En-Ciel 2000” (project no. 8). 1 Under isotopy, as usual, we understand a smooth isotopy in the real part of the ambient algebraic surface. c 2003 Independent University of Moscow 1053