Real plane separating (M-2)-curves of degree d and totally real pencils of degree d-3

TL;DR

The paper generalizes the classical quintic result that a real plane curve is separating if and only if its ovals are in non-convex position to all real plane separating $(M-2)$-curves of degree $d\ge 4$. It proves that such curves admit infinitely many totally real pencils of degree $d-3$, with the number of base points determined by the separating gonality: $g-1$ base points when sepgon$(C)=g-1$ and $g-2$ base points when sepgon$(C)=g$, where $g=\frac{(d-1)(d-2)}{2}$ is the genus. The approach combines Gabard’s bound on separating gonality, constructions via separating morphisms, and Bezout-type intersection arguments, with supplementary analysis of complex orientations and the separating semigroup Sep$(C)$. A quintic example ($C_5$) is used to illustrate constraints on separating gonality and to discuss implications for the semigroup and orientation data, highlighting the broader applicability to real plane curves. Overall, the work provides a constructive path to totally real pencils on separating $(M-2)$-curves and deepens connections between real algebraic geometry, morphism theory, and complex orientations.

Abstract

It is well known that a non-singular real plane projective curve of degree five with five connected components is separating if and only if its ovals are in non-convex position. In this article, this property is set into a different context and generalised to all real plane separating (M-2)-curves.

Figures (3)

• Figure 1: Arrangement of a triplet $(\mathbb P^2 (\mathbb R), C(\mathbb R), S_1 \cup S_2 \cup S_3)$ as in Definition \ref{['defn: non-convex']}, where the $S_i$ are the three segments.
• Figure 2: $(\mathbb P^2 (\mathbb R) \setminus J, C_5(\mathbb R), L_0(\mathbb R) \cup L_1(\mathbb R))$. The arrows denote the fixed complex orientation of the ovals of $C(\mathbb R)$, and the dots $\bullet$ denote the points of tangency and the intersection point of the fixed lines $L_0$ and $L_1$, which are in bold.
• Figure 3: $(\mathbb P^2 (\mathbb R), C(\mathbb R), L(\mathbb R))$ of Example \ref{['exa: quintic']}. Double arrows denote $\mathfrak O$, simple arrows the fixed complex orientation of $C(\mathbb R)$ and $\bullet$ the points in $f^{-1}(p)$.

Theorems & Definitions (21)

• Definition 1.1
• Lemma 1.2
• Definition 1.3
• Definition 1.4
• Remark 1.5
• Definition 1.6
• Lemma 1.7: cf. Fied83
• Lemma 1.8: Special case of the complex orientations formula of Mish75
• proof : Proof of \ref{['lem: quintic']}
• Theorem 1.9