  We study the period map of configurations of n points on the projective line
constructed via a cyclic cover branching along these points. By considering the
decomposition of its Hodge structure into eigenspaces, we establish the
codimension of the locus where the eigenperiod map is still pure. Furthermore,
we show that the period map extends to the divisors of a specific moduli space
of weighted stable rational curves, and that this extension satisfies a local
Torelli map along its fibers.
