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The GIT of 4x3x3 tensors (determinantal cubic surfaces)

  • Charlotte Sherratt
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Abstract

A general 4×3×3 tensor defines three determinantal varieties in ℙ3 and ℙ2 and ℙ2 where it drops rank: a cubic surface and two length 6 subschemes. We study the Geometric Invariant Theory (GIT) quotient of ℙ(ℂ4⊗ℂ3⊗ℂ3) by SL(4)×SL(3)×SL(3), a moduli space of linear determinantal representations of cubic surfaces. We give a geometric description of the (semi)stable locus and some of the geometry of this GIT quotient. This GIT quotient admits a surjective rational map to the GIT moduli space of cubic surfaces, undefined at only one point.

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Cite

BibTeXRIS
BibTeX
@misc{sherratt20264x3x3,
  title = {{The GIT of 4x3x3 tensors (determinantal cubic surfaces)}},
  author = {Charlotte Sherratt},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28497},
  eprint = {2609.28497},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28497},
}
RIS
TY  - UNPB
TI  - The GIT of 4x3x3 tensors (determinantal cubic surfaces)
AU  - Charlotte Sherratt
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.28497
UR  - https://arxiv.org/abs/2609.28497
AB  - A general 4×3×3 tensor defines three determinantal varieties in ℙ^3 and ℙ^2 and ℙ^2 where it drops rank: a cubic surface and two length 6 subschemes. We study the Geometric Invariant Theory (GIT) quotient of ℙ(ℂ^4⊗ℂ^3⊗ℂ^3) by SL(4)×SL(3)×SL(3), a moduli space of linear determinantal representations of cubic surfaces. We give a geometric description of the (semi)stable locus and some of the geometry of this GIT quotient. This GIT quotient admits a surjective rational map to the GIT moduli space of cubic surfaces, undefined at only one point.
ER  -