Relative geometric invariant theory
type: Item
instance of: scholarly article
title: Relative geometric invariant theory
zbMATH Open document ID: 1482.14047
DOI: 10.4171/LEM/1010
author: Alexander Schmitt
published in: L'Enseignement Mathématique. 2e Série
publication date: 2022-02-24T00:00:00Z
review text: This paper provides new proofs for certain results on relative geometric invariant theory (relative GIT). GIT explores the structure of the quotient of an algebraic variety \(X\) by an algebraic group \(G\) and dates back to Hilbert who shows that if \(X\) is complex affine and \(G\) is complex reductive linear, the ring of invariants \(\mathbb{C}[X]^G\) is finitely generated, then we can define the quotient of \(X\) by \(G\) as the affine variety \(X/\!\!/ G=Spec(\mathbb{C}[V]^G)\). Mumford extends these notions to a more general setting, when \(X\) is projective, posing the notion of stability to remove certain unstable orbits in the quotient, measured by the so-called Hilbert-Mumford criterion. And also introducing the notion of linealization of the action to a line bundle to find affine \(G\)-invariant subsets of \(X\) to glue together in order to get a quotient of \(X\) by \(G\). The relative setting of this is given by a reductive linear algebraic group \(G\), quasi-projective algebraic varieties \(X\), \(Y\) with actions of \(G\), and a \(G\)-equivariant projective morphism \(\pi:X\rightarrow Y\). The question is whether there exists, given a linearization on \(Y\), a linearization on \(X\) relating the semistable points in both GIT quotients inducing a projective morphism \(\overline{\pi}:X/\!\!/ G \rightarrow Y/\!\!/ G\). When \(X\) and \(Y\) are projective, \textit{Z. Reichstein} [Invent. Math. 96, No. 2, 349--383 (1989; Zbl 0675.14021)] shows a positive solution for this problem, and Theorem 3.3 provides a simpler proof of this fact, based on a study of the minimal weights appearing in the Hilbert-Mumford criterion to measure stability when moving on the cone of one-parameter subgroups. \textit{L. H. Halle} et al. [``Relative VGIT and an application to degenerations of Hilbert schemes'', Preprint, \url{arXiv:1909.03780}] study this problem when \(Y\) is affine. The above argument is adapted to give an alternative proof of their result (c.f. Theorem 4.1) s…
reviewed by: Alfonso Zamora
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14D23
zbMATH DE Number: 7480962
zbMATH Keywords: quotient
zbMATH Keywords: semistability
zbMATH Keywords: Hilbert-Mumford criterion
zbMATH Keywords: equivariant morphism
zbMATH Keywords: variation of quotients and moduli spaces
zbMATH Keywords: relative moduli space
MaRDI profile type: Publication
cites work: A functorial construction of moduli of sheaves
cites work: Quivers, geometric invariant theory, and moduli of linear dynamical systems
cites work: Finiteness of the number of maximal open subsets with good quotients
cites work: Q3999894
cites work: Unzerlegbare Darstellungen. I. (Indecomposable representations. I)
cites work: Lectures on moduli of curves. Notes by D. R. Gokhale
cites work: Q5322842
cites work: A relative Hilbert-Mumford criterion
cites work: A GIT construction of degenerations of Hilbert schemes of points
cites work: Q4143433
cites work: A wild quiver in linear systems theory
cites work: Infinite root systems, representations of graphs and invariant theory
cites work: Desingularizations of the moduli space of rank 2 bundles over a curve
cites work: MODULI OF REPRESENTATIONS OF FINITE DIMENSIONAL ALGEBRAS
cites work: MODULI OF BRILL-NOETHER PAIRS ON ALGEBRAIC CURVES
cites work: Q3346437
cites work: Partial desingularisations of quotients of nonsingular varieties and their Betti numbers
cites work: On the Homology of Compactifications of Moduli Spaces of Vector Bundles Over a Riemann Surface
cites work: Good degeneration of Quot-schemes and coherent systems
cites work: Q5534729
cites work: Geometric Invariant Theory
cites work: Q3463652
cites work: A Compactification over \overline{𝑀_{𝑔}} of the Universal Moduli Space of Slope-Semistable Vector Bundles
cites work: Moduli of pairs and generalized theta divisors
cites work: Stability and equivariant maps
cites work: The GIT-equivalence for \(G\)-line bundles
cites work: A simple proof for the finiteness of GIT-quotients
cites work: A universal construction for moduli spaces of decorated vector bundles over curves
cites work: The Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves
cites work: Geometric invariant theory and decorated principal bundles
cites work: Q2966312
cites work: Generically semistable linear quiver sheaves
cites work: A remark on relative geometric invariant theory for quasi‐projective varieties
cites work: Notes on coherent systems
full work available at URL: https://doi.org/10.4171/lem/1010
OpenAlex ID: W3217346395
description: scientific article; zbMATH DE number 7480962