  We consider the family of constant curvature fiber metrics for a Lefschetz
fibration with regular fibers of genus greater than one. A result of Obitsu and
Wolpert is refined by showing that on an appropriate resolution of the total
space, constructed by iterated blow-up, this family is log-smooth, i.e.
polyhomogeneous with integral powers but possible multiplicities, at the
preimage of the singular fibers in terms of parameters of size comparable to
the logarithm of the length of the shrinking geodesic.
