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Geometric Invariant Theory for Affine Superschemes

  • Alexander Quintero Velez
  • Pedro Rizzo
  • Alexander Torres-Gomez
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Abstract

We develop Geometric Invariant Theory for affine superschemes under the action of reductive algebraic supergroups, formulating the theory in terms of coordinate Hopf superalgebras in order to accommodate anticommuting and nilpotent variables. Within this algebraic setting, we construct affine superquotients and establish their basic structural properties under suitable hypotheses. We then prove a supergeometric analogue of the Hilbert-Mumford criterion, giving a numerical test for the semistability and stability of points under natural constraints on the supergroup coaction. We show that this numerical criterion can be used to construct the GIT superquotient of these superschemes, and we give examples illustrating how the supergeometric GIT superquotient differs from its classical counterpart.

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Cite

BibTeXRIS
BibTeX
@misc{velez2026geometric,
  title = {{Geometric Invariant Theory for Affine Superschemes}},
  author = {Alexander Quintero Velez and Pedro Rizzo and Alexander Torres-Gomez},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14904},
  eprint = {2609.14904},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14904},
}
RIS
TY  - UNPB
TI  - Geometric Invariant Theory for Affine Superschemes
AU  - Alexander Quintero Velez
AU  - Pedro Rizzo
AU  - Alexander Torres-Gomez
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.14904
UR  - https://arxiv.org/abs/2609.14904
AB  - We develop Geometric Invariant Theory for affine superschemes under the action of reductive algebraic supergroups, formulating the theory in terms of coordinate Hopf superalgebras in order to accommodate anticommuting and nilpotent variables. Within this algebraic setting, we construct affine superquotients and establish their basic structural properties under suitable hypotheses. We then prove a supergeometric analogue of the Hilbert-Mumford criterion, giving a numerical test for the semistability and stability of points under natural constraints on the supergroup coaction. We show that this numerical criterion can be used to construct the GIT superquotient of these superschemes, and we give examples illustrating how the supergeometric GIT superquotient differs from its classical counterpart.
ER  -