Special Cohen--Macaulay sheaves on partial resolutions of rational surfaces singularities

@misc{uehara2026special,
  title = {{Special Cohen--Macaulay sheaves on partial resolutions of rational surfaces singularities}},
  author = {Hokuto Uehara},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30645},
  eprint = {2609.30645},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30645},
}

TY  - UNPB
TI  - Special Cohen--Macaulay sheaves on partial resolutions of rational surfaces singularities
AU  - Hokuto Uehara
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30645
UR  - https://arxiv.org/abs/2609.30645
AB  - We introduce the categories CM(X) and SCM(X) of reflexive sheaves on a minimal partial resolution f: X → Spec R of a rational surface singularity Spec R. The main result of this paper establishes that SCM(X) possesses a natural Frobenius structure, serving as a geometric counterpart to the algebraic Frobenius structure on special Cohen--Macaulay R-modules, introduced by Iyama--Wemyss and Iyama--Kalck--Wemyss--Yang. Utilizing this geometric framework, we establish an exact equivalence between SCM(X) and the category of special Cohen--Macaulay R-modules equipped with a specific exact structure, which induces a triangle equivalence between their stable categories. Consequently, this provides a direct, geometric proof of the Iyama--Kalck--Wemyss--Yang equivalence and yields a Buchweitz-type equivalence underlineSCM(X) ≃ D_sg(X).
ER  -