Obstructions for the existence of separating morphisms and totally real pencils

Abstract

It goes back to Ahlfors that a real algebraic curve $C$ admits a separating morphism $f$ to the complex projective line if and only if the real part of the curve disconnects its complex part, i.e. the curve is \textit{separating}. The degree of such $f$ is bounded from below by the number $l$ of real connected components of $ \mathbb{R} C$. The sharpness of this bound is not a priori clear. We prove that real algebraic separating curves, embedded in some ambient surface and with $l$ bounded in a certain way, do not admit separating morphisms of lowest possible degree. Moreover, this result of non-existence can be applied to show that certain real separating plane curves of degree $d$, do not admit totally real pencils of curves of degree $k$ such that $kd \leq l$.