  We offer a groupoid-theoretic approach to computing invariants. We illustrate
this approach by describing the Gel'fand-MacPherson correspondence and the Gale
transform as well as giving Zariski-local descriptions of the moduli space of
ordered points in P^1. We give an explicit description of the moduli space
M_0(P^1,2) over Spec Z. In characteristic 2, there is a singularity at the
totally ramified cover which is isomorphic to the affine cone over the Veronese
embedding P^1 --> P^4.
