Pencils on separating (M-2)-curves

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 $g\geq 4$.