Pencils on separating (M-2)-curves

TL;DR

This paper explores the structure of separating (M-2)-curves, relating partitions of their Teichmüller space via special types and separating gonality, and demonstrates the existence of certain real curves with specific linear systems.

Contribution

It establishes a close relationship between two partitions of the Teichmüller space of separating (M-2)-curves and proves the existence of real curves with isolated real linear systems for all genera g ≥ 4.

Findings

Two partitions of Teichmüller space are closely related.

Existence of real curves with isolated real linear systems g^1_{g-1} for all g ≥ 4.

Connection between special types and separating gonality of curves.

Abstract

A separating ($M-2$)-curve is a smooth geometrically irreducible real projective curve $X$ such that $X(\mathbb{R})$ has $g-1$ connected components and $X(\mathbb{C})\setminus X(\mathbb{R})$ is disconnected. Let $T_g$ be a Teichm\"uller space of separating ($M-2$)-curves of genus $g$. We consider two partitions of $T_g$, one by means of a concept of special type, the other one by means of the separating gonality. We show that those two partitions are very closely related to each other. As an application we obtain the existence of real curves having isolated real linear systems $g^1_{g-1}$ for all $g\geq 4$.