Obstructions for the existence of separating morphisms and totally real pencils

TL;DR

This paper investigates conditions under which real algebraic curves admit separating morphisms and totally real pencils, establishing obstructions based on the topology of the real part and degree bounds.

Contribution

It proves non-existence results for separating morphisms and totally real pencils under specific topological and degree constraints.

Findings

Certain real separating curves do not admit separating morphisms of minimal degree.

Obstructions to totally real pencils are established for specific degree and topological configurations.

Results extend understanding of real algebraic curves' morphisms and 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 $kd \leq l$.