Hypergraphs associated to skew polynomial algebras and surface triangulations
Abstract
We study the realization problem for point schemes of skew polynomial algebras. We approach this problem combinatorially by translating it into the study of certain $3$-uniform hypergraphs, which we call point hypergraphs. We first give a homological criterion characterizing point hypergraphs. Using this criterion, we study hypergraphs obtained by deleting triangles from triangulations of connected orientable closed surfaces. Deleting exactly one triangle always gives a non-point hypergraph. When the triangulation has no separating nonfacial $3$-cycles, deleting either no triangles or at least two triangles gives a point hypergraph, and the one-triangle deletions are minimal non-point hypergraphs with respect to taking induced sub-hypergraphs. It follows that point hypergraphs cannot be characterized by finitely many forbidden induced sub-hypergraphs. Next, for each point hypergraph, we construct an affine moduli variety of skew polynomial algebras realizing it, and determine its dimension. For point hypergraphs arising from the above surface construction, we obtain an explicit dimension formula in terms of the number of vertices, the Euler characteristic, and the number of deleted triangles. Finally, we characterize point hypergraphs on six vertices in terms of a four-vertex local condition together with a single exceptional obstruction.
Related
- Polynomial-in-r bounds for forbidden traces of uniform hypergraphs
- Filtered deformations of three-variable polynomial algebras
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- On the Isotropy Groups of Locally Finite Derivations of Laurent Polynomial Algebras
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BibTeX
@misc{higashitani2026hypergraphs,
title = {{Hypergraphs associated to skew polynomial algebras and surface triangulations}},
author = {Akihiro Higashitani and Kenta Ueyama},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05311},
eprint = {2610.05311},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05311},
}
RIS
TY - UNPB
TI - Hypergraphs associated to skew polynomial algebras and surface triangulations
AU - Akihiro Higashitani
AU - Kenta Ueyama
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05311
UR - https://arxiv.org/abs/2610.05311
AB - We study the realization problem for point schemes of skew polynomial algebras. We approach this problem combinatorially by translating it into the study of certain $3$-uniform hypergraphs, which we call point hypergraphs. We first give a homological criterion characterizing point hypergraphs. Using this criterion, we study hypergraphs obtained by deleting triangles from triangulations of connected orientable closed surfaces. Deleting exactly one triangle always gives a non-point hypergraph. When the triangulation has no separating nonfacial $3$-cycles, deleting either no triangles or at least two triangles gives a point hypergraph, and the one-triangle deletions are minimal non-point hypergraphs with respect to taking induced sub-hypergraphs. It follows that point hypergraphs cannot be characterized by finitely many forbidden induced sub-hypergraphs. Next, for each point hypergraph, we construct an affine moduli variety of skew polynomial algebras realizing it, and determine its dimension. For point hypergraphs arising from the above surface construction, we obtain an explicit dimension formula in terms of the number of vertices, the Euler characteristic, and the number of deleted triangles. Finally, we characterize point hypergraphs on six vertices in terms of a four-vertex local condition together with a single exceptional obstruction.
ER -