by  S.Yu. Orevkov

Abstract:  We prove non-realizability of 2 complex M-schemes of plane real algebraic curves degree 7, and the realizability of some other ones. For the prohibitions we apply Murasugi-Tristram inequality for an auxilary link determined by a pencil of lines. We improve this method using vanishing cycles coming from other pencils of lines. In Appendix 1 we apply these methods to obtain restrictions for 8th degree M-curves <4 U 1<2 U 1<14>> and <14 U 1<2 U 1<4>> (their realizability is still unknown). We realize also these schemes by pseudo-holomorphic curves. In Appendix 2 we illustrate how to use the Goeritz matrix. We prove non-existence of certain (M-1)-perturbation of four tangent branches.