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Symmetric plane curves of degree 7 : pseudo-holomorphic and algebraic classifications

Almost all known restrictions on the topology of nonsingular real algebraic curves in the projective plane are also valid for a wider class of objects: real pseudo-holomorphic curves. It is still unknown if there exists a nonsingular real pseudo-holomorphic curve not isotopic in the projective plane to a real algebraic curve of the same degree. In this article, we focus our study on symmetric real curves in the projective plane. We give a classification of real schemes (resp. complex schemes) realizable by symmetric real curves of degree 7 with respect to the type of the curve (resp. M-symmetric real curves of degree 7). In particular, we exhibit two real schemes which are realizable by real symmetric dividing pseudo-holomorphic curves of degree 7 in the projective plane but not by algebraic ones. The algebraic prohibitions are proved using tools introduced by S. Orevkov: the comb theoretical method and the cubic resolvent.