  We describe a relation between the invariants of $n$ ordered points in $P^d$
and of points contained in a union of linear subspaces $P^{d1}\cup P^{d2}
\subset P^d$. This yields an attaching map for GIT quotients parameterizing
point configurations in these spaces, and we show that it respects the Segre
product of the natural GIT polarizations. Associated to a configuration
supported on a rational normal curve is a cyclic cover, and we show that if the
branch points are weighted by the GIT linearization and the rational normal
curve degenerates, then the admissible covers limit is a cyclic cover with
weights as in this attaching map. We find that both GIT polarizations and the
Hodge class for families of cyclic covers yield line bundles on $\bar{M}_{0,n}$
with functorial restriction to the boundary. We introduce a notion of
divisorial factorization, abstracting an axiom from rational conformal field
theory, to encode this property and show that it determines the isomorphism
class of these line bundles. As an application, we obtain a unified, geometric
proof of two recent results on conformal block bundles, one by Fedorchuk and
one by Gibney and the second author.
