Computing invariants via slicing groupoids: Gel'fand MacPherson, Gale and positive characteristic stable maps

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.