Computing invariants via slicing groupoids: Gel'fand MacPherson, Gale and positive characteristic stable maps
type: Item
instance of: scholarly article
title: Computing invariants via slicing groupoids: Gel'fand MacPherson, Gale and positive characteristic stable maps
zbMATH Open document ID: 1262.13009
published in: Mathematische Nachrichten
publication date: 2012-05-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.3448
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 14D23
zbMATH DE Number: 6034376
zbMATH Keywords: Invariant theory
zbMATH Keywords: groupoids
zbMATH Keywords: Artin stacks
author: Jarod Alper
MaRDI profile type: Publication
OpenAlex ID: W2121516952
arXiv ID: 1011.3448
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DOI: 10.1002/MANA.201000058
review text: 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.
reviewed by: Gerhard Pfister
description: scientific article; zbMATH DE number 6034376