by S.Yu. Orevkov and E.I. Shustin

In the present paper we classify real plane curves of degree 8 with a singular point of quadratic tangency of four smooth branches (equivalently, curves of degree 8 on the quadratic cone, or curves of bidegree (8,4) of the Hirzebruch surface $F_2$). Our main result consists in the two isotopy classifications, a classification of real algebraic curves, and a classification of real pseudo-holomorphic curves, which show that there exist real pseudo-holomorphic curves which are not isotopic to any real algebraic curve (in the corresponding class). Real pseudo-holomorphic curves form a subclass of flexible curves introduced by Viro. He also posed a problem to distinguish between flexible and algebraic curves, which we answer in the considered case.

The pseudo-holomorphic classification is based on the braid group technique developed by the first author and Gromov's theory of pseudo-holomorphic curves. To prohibit non-algebraic pseudo-holomorphic curves we apply the Hilbert-Rohn-Gudkov method.