construct Seifert fibration on mapping torus of surface with monodromy a periodic mapping class

Question

How do we construct Seifert fibration on mapping torus of surface with monodromy a periodic mapping class. I know that the fiber of the Seifert fibration has to be transverse to the surface fiber of the fibration over $S^1$ but I do not know how we do the construction. Under which conditions can we do the same construction for a reducible mapping class ?

I have another independent question: for a reducible mapping class, do the invariant multicurve corresponds to the set of splitting tori of the JSJ-decomposition.

Answer 1

Just use the suspension flow of the periodic diffeomorphism $f: S\to S$ in the periodic mapping class. Then all flow lines will be periodic (i.e., circles) and you are done; the base will be the quotient $S/f$.

For the second question, the answer is yes; again, just suspend the invariant multicurve.