2015/12/09 by Gergely Bérczi, Bérczi, Gergely, Brent Doran +5
Computer Science · Mathematics · #14L24 #14L30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1512.02997
openalex publication_date 2015/12/09 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
In this article we review the question of constructing geometric quotients of\nactions of linear algebraic groups on irreducible varieties over algebraically\nclosed fields of characteristic zero, in the spirit of Mumford's geometric\ninvariant theory (GIT). The article surveys some recent work on geometric\ninvariant theory and quotients of varieties by linear algebraic group actions,\nas well as background material on linear algebraic groups, Mumford's GIT and\nsome of the challenges that the non-reductive setting presents. The earlier\nwork of two of the authors in the setting of unipotent group actions is\nextended to deal with actions of any linear algebraic group. Given the data of\na linearisation for an action of a linear algebraic group H on an irreducible\nvariety X, an open subset of stable points Xs is defined which admits a\ngeometric quotient variety Xs/H. We construct projective completions of the\nquotient Xs/H by considering a suitable extension of the group action to an\naction of a reductive group on a reductive envelope and using Mumford's GIT. In\ngood cases one can also compute the stable locus Xs in terms of stability\n(in the sense of Mumford for reductive groups) for the reductive envelope.\n