 
Summary: COMPUTING INVARIANTS VIA SLICING:
GEL'FAND MACPHERSON, GALE AND POSITIVE CHARACTERISTIC KONTSEVICH
JAROD ALPER
Abstract. We offer a groupoidtheoretic approach to computing invariants. We illustrate this
approach by describing the Gel'fandMacPherson correspondence and the Gale transform. We also
provide Zariskilocal descriptions of the moduli space of ordered points in P1. We give an explicit
description of the moduli space M0(P1, 2) over Spec Z. In characteristic 2, the singularity at the
totally ramified cover is isomorphic to the affine cone over the Veronese embedding P1 P4.
1. Introduction
The central question in classical invariant theory is to describe the graded ring of invariants
n0(X, O(n))G
where G is an algebraic group acting linearly on a projective space P(V ) and
X P(V ) is a Ginvariant subvariety. In this paper, we show that one can sometimes "slice" the
groupoid G × X X by a subvariety W X which is suitably transverse to the generic orbit to
produce a groupoid RW W (not necessarily arising from a group action) where it is easier to
compute the invariants. Specifically, suppose X is normal and g : W X is a finite type morphism
such that the composition G × W G × X
 X is flat whose image G · W X has a complement
of codimension at least 2, then RW := G × X ×X×X W × W W is a flat groupoid and there is a
