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PreprintCC BY 4.0arXiv

Point Counts of Cluster Varieties of Marked Surfaces Over Finite Fields

  • James Beyer
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Abstract

We establish formulae for point counts of cluster varieties of cluster algebras of marked surfaces, possibly with punctures. We then establish formulae for the number of non-deep points in these cluster varieties over 𝔽2, which gives us the number of algebraic tori necessary to cover the cluster manifold over 𝔽2. We also show that these formulae satisfy certain recurrence relations.

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Cite

BibTeXRIS
BibTeX
@misc{beyer2026point,
  title = {{Point Counts of Cluster Varieties of Marked Surfaces Over Finite Fields}},
  author = {James Beyer},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.22208},
  eprint = {2608.22208},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.22208},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Point Counts of Cluster Varieties of Marked Surfaces Over Finite Fields
AU  - James Beyer
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2608.22208
UR  - https://arxiv.org/abs/2608.22208
AB  - We establish formulae for point counts of cluster varieties of cluster algebras of marked surfaces, possibly with punctures. We then establish formulae for the number of non-deep points in these cluster varieties over 𝔽_2, which gives us the number of algebraic tori necessary to cover the cluster manifold over 𝔽_2. We also show that these formulae satisfy certain recurrence relations.
ER  -