The vanishing cycles of curves in toric surfaces II

We resume the study initiated in \cite{CL}. For a generic curve $C$ in an ample linear system $\vert \mathcal{L} \vert$ on a toric surface $X$, a vanishing cycle of $C$ is an isotopy class of simple closed curve that can be contracted to a point along a degeneration of $C$ to a nodal curve in $\vert \mathcal{L} \vert$. The obstructions that prevent a simple closed curve in $C$ from being a vanishing cycle are encoded by the adjoint line bundle $K_X \otimes \mathcal{L}$. In this paper, we consider the linear systems carrying the two simplest types of obstruction. Geometrically, these obstructions manifest on $C$ respectively as an hyperelliptic involution and as a Spin structure. In both cases, we determine all the vanishing cycles by investigating the associated monodromy maps, whose target space is the mapping class group $MCG(C)$. We show that the image of the monodromy is the subgroup of $MCG(C)$ preserving respectively the hyperelliptic involution and the Spin structure. In particular, we provide an explicit finite set of generators for the Spin mapping class group. The results obtained here support the Conjecture $1$ in \cite{CL} aiming to describe all the vanishing cycles for any pair $(X, \mathcal{L})$.