Computing invariants via slicing groupoids: Gel'fand MacPherson, Gale and positive characteristic stable maps (Q2883879)

Let \(G\) be an algebraic group acting linearly on a projective space \(\mathbb{P}(V)\) and \(X\subseteq \mathbb{P}(V)\) a \(G\)--invariant subvariety. A groupoid--theoretic approach to compute invariants is proposed by slicing the groupoid \(G\times X\rightrightarrows X\) by a subvariety \(W\subset X\) which is suitably transverse to the generic orbit to produce a groupoid \(R|_W\rightrightarrows W\) where it is easier to compute invariants. This idea is illustrated giving generalizations of the classical Gelfand-MacPherson correspondence and Gale transform. The slicing technique is used to give Zariski--local descriptions of the moduli space of \(n\) order points in \(\mathbb{P}^1\). Finally a global description of the Kontsevich moduli space of stable maps \(M_0(\mathbb{P}^1, 2)\) (the coarse moduli space parametrizing non--constant, degree 2 morphisms \(\mathbb{P}^1\to \mathbb{P}^1\) modulo automorphisms of the source) is given and its GIT compactification.