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PreprintCC BY 4.0arXiv

Relative Geometric Invariant Theory: Reductive and Non-reductive

  • Eloise Hamilton
  • Victoria Hoskins
  • Joshua Jackson
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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.

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Cite

BibTeXRIS
BibTeX
@misc{hamilton2026relative,
  title = {{Relative Geometric Invariant Theory: Reductive and Non-reductive}},
  author = {Eloise Hamilton and Victoria Hoskins and Joshua Jackson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05942},
  eprint = {2609.05942},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05942},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Relative Geometric Invariant Theory: Reductive and Non-reductive
AU  - Eloise Hamilton
AU  - Victoria Hoskins
AU  - Joshua Jackson
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.05942
UR  - https://arxiv.org/abs/2609.05942
AB  - 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.
ER  -