Pseudoholomorphic Algebraically Unrealizable Curves

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. 2000 Math. Subj. Class. Primary 14P25, 57M25; Secondary 14H20, 53D99.