Obstructions for the existence of separating morphisms and totally real pencils

It goes back to Ahlfors that a real algebraic curve <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> admits a separating morphism <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> is bounded from below by the number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>l</mml:mi> </mml:math> of real connected components of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℝ</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> . The sharpness of this bound is not a priori clear. We prove that real algebraic separating curves, embedded in some ambient surface and with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>l</mml:mi> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> , do not admit totally real pencils of curves of degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>l</mml:mi> </mml:mrow> </mml:math> .