HEXAGONbeta

DEEP LOCI OF DOUBLE BOTT–SAMELSON VARIETIES

A mix of human-written and AI-generated textHuman understanding: some partsmath.SG — Symplectic Geometrymath.AG — Algebraic Geometry

Contributed by Qiyue Tang ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

We prove that the deep locus of a complex double Bott–Samelson variety associated with a positive braid is its stabilizer locus. Consequently, the deep locus is empty if and only if any two components of the braid closure are unlinked. This resolves Conjecture 4.45 of Castronovo, Gorsky, Simental, and Speyer.

Provenance statement

The proof was developed in a ChatGPT-6-Pro research conversation whose recorded continuation lasted approximately 480 minutes, building on earlier project materials. The research included AI dialogue review and exact computations. Codex subsequently audited the arguments and prepared and revised the English manuscript. An earlier version passed an AI review using Rethlas-OSV , based on the OSVerify framework of Nie et al.

References

  1. A. Björner and F. Brenti, Combinatorics of Coxeter Groups , Graduate Texts in Mathematics 231, Springer, 2005. doi:10.1007/3-540-27596-7 (https://doi.org/10.1007/3-540-27596-7).DOI
  2. J. Beyer and G. Muller, Deep Points of Cluster Algebras , arXiv:2403.15589v1 (https://arxiv.org/abs/2403.15589v1), 2024.arXiv
  3. R. Casals, E. Gorsky, M. Gorsky, I. Le, L. Shen, and J. Simental, Cluster structures on braid varieties , arXiv:2207.11607v2 (https://arxiv.org/abs/2207.11607v2), 2024.arXiv
  4. M. Castronovo, M. Gorsky, J. Simental, and D. E. Speyer, Cluster deep loci and mirror symmetry , arXiv:2402.16970v2 (https://arxiv.org/abs/2402.16970v2), 2024.arXiv
  5. S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations , J. Amer. Math. Soc. 15 (2002), 497--529; arXiv:math/0104151v1 (https://arxiv.org/abs/math/0104151v1).arXiv
  6. H. Gao, L. Shen, and D. Weng, Augmentations, Fillings, and Clusters , arXiv:2008.10793v2 (https://arxiv.org/abs/2008.10793v2), 2024.arXiv
  7. E. Gorsky, S. Kim, T. Scroggin, and J. Simental, Splicing braid varieties , Canad. Math. Commun. 1 (2026), e4, 1--48; doi:10.4153/S2976859426100034 (https://doi.org/10.4153/S2976859426100034).DOI
  8. X. He and S. Nie, Minimal length elements of finite Coxeter groups , Duke Math. J. 161 (2012), 2945--2967; arXiv:1108.0282v1 (https://arxiv.org/abs/1108.0282v1).arXiv
  9. G. Muller, Locally acyclic cluster algebras , Adv. Math. 233 (2013), 207--247; arXiv:1111.4468v1 (https://arxiv.org/abs/1111.4468v1).arXiv
  10. S. Neville and J. Simental, Mysterious points in keys but not trees , arXiv:2603.17890v1 (https://arxiv.org/abs/2603.17890v1), 2026.arXiv
Show all 15 referencesShow fewer references
  1. F. Nie, H. Ye, S. Liu, Z. Wang, S. Ermon, and J. Zou, One-Sided Verification Under Limited Expert Review , arXiv:2610.18089v2 (https://arxiv.org/abs/2610.18089v2), 2026.arXiv
  2. OSVerify contributors, Rethlas-OSV , software, https://github.com/OSVerify/Rethlas-OSV, accessed November 1, 2026.link
  3. H. Quan, On cluster deep loci in double Bruhat cells , arXiv:2608.27870v1 (https://arxiv.org/abs/2608.27870v1), 2026.arXiv
  4. L. Shen and D. Weng, Cluster Structures on Double Bott--Samelson Cells , arXiv:1904.07992v3 (https://arxiv.org/abs/1904.07992v3), 2022.arXiv
  5. M. J. Tyler, Locally Acyclic Surface Type and Finite Type Cluster Algebras Have No Mysterious Points , arXiv:2608.21272v1 (https://arxiv.org/abs/2608.21272v1), 2026.arXiv

Version history

  1. v1 · 15 pagesSubmitted by Qiyue TangInitial depositCurrentNov 02, 2026