Obstructions for the existence of separating morphisms and totally real pencils

Manzaroli, Matilde

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 5, pp. 1233-1250

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 separating. The degree of such $f$ is bounded from below by the number $l$ of real connected components of $ℝC$. 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 $kd\le l$.

Il remonte à Ahlfors qu’une courbe algébrique réelle $C$ admet un morphisme séparant $f$ à la droite complexe projective si et seulement si la partie réelle de la courbe déconnecte sa partie complexe, i.e. la courbe est séparante. Le degré d’un tel $f$ est borné par en dessous par le nombre de composantes connexes réelles de $ℝC$. L’optimalité de cette borne n’est pas claire a priori. Nous prouvons que les courbes algébriques réelles séparantes, plongées dans une surface ambiante et avec $l$ borné d’une certaine manière, n’admettent pas de morphismes séparants de degré le plus petit possible. De plus, ce résultat de non-existence peut être appliqué pour montrer que certaines courbes réelles séparantes planes de degré $d$, n’admettent pas de pinceaux de courbes de degré $k$ totalement réels tels que $kd\le l$.

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