Suppose we have a Calabi-Yau 3-fold $X$ (not necessarily compact, over $\mathbb{C}$) that contains a ruled surface over a smooth curve $C$ of genus $g$. I am using a strong definition of a ruled surface, meaning that we have a fibration over $C$ where all the fibers are $\mathbb{P}^1$. If a fiber can be contracted in $X$, then all the other fibers must contract as well, and we get transverse $A_1$ singularities along $C$.
My question is, what can we say about the singularity, when we have a reducible fiber over a point? For instance, suppose we have a surface $S$ obtained as a blow up of a point on a fiber of a Hirzebruch surface and $X=\text{Tot }K_S$, the total space of the canonical bundle. Then there is a map $X\to Y$ to a singular Calabi-Yau 3-fold $Y$ collapsing all the fibers (viewing $S$ as a ruled surface) of $S$. Then on $Y$, we have a generically transverse $A_1$ singularity and a dissident point. What is the singularity at this dissident point?
