{
  "abstract": "We construct good quotients for equivariant actions of group homomorphisms on morphisms of schemes. Using Geometric Invariant Theory, we obtain explicit open semistable loci in the source with Hilbert-Mumford descriptions admitting good quotients relative to a given good quotient of the target. In particular, we obtain quotients for reductive groups acting on projective-over-affine morphism. In the non-reductive case, we consider equivariant actions on affine morphisms which are 'graded' by a multiplicative group and satisfy certain unipotent stabiliser assumptions. This recovers known results in projective non-reductive GIT as a special case, which also proves the Hilbert-Mumford criterion in that setting. As applications, we consider moduli of unstable objects, representations of quivers with multiplicities and jets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05942",
  "author_total": 3,
  "authors": [
    {
      "name": "Eloise Hamilton"
    },
    {
      "name": "Victoria Hoskins"
    },
    {
      "name": "Joshua Jackson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3zd1amgqrz0d3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05942",
  "fulltext_url": null,
  "id": "3zd1amgqrz0d3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05942",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7212077611",
  "pdf_url": "https://arxiv.org/pdf/2609.05942",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3zd1amgqrz0d3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05942",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05942"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Relative Geometric Invariant Theory: Reductive and Non-reductive",
  "title_text": "Relative Geometric Invariant Theory: Reductive and Non-reductive",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/3zd1amgqrz0d3",
  "venue": "arXiv",
  "versions": []
}