Iyama, O. and Wemyss, M. (2020) Weighted projective lines and rational surface singularities. Épijournal de Géométrie Algébrique, 3, 13.
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Abstract
In this paper we study rational surface singularities R with star shaped dual graphs, and under very mild assumptions on the self-intersection numbers we give an explicit description of all their special Cohen-Macaulay modules. We do this by realising R as a certain Z-graded Veronese subring S^x of the homogeneous coordinate ring S of the Geigle-Lenzing weighted projective line X, and we realise the special CM modules as explicitly described summands of the canonical tilting bundle on X. We then give a second proof that these are special CM modules by comparing qgr S^x and coh X, and we also give a necessary and sufficient combinatorial criterion for these to be equivalent categories. In turn, we show that qgr S^x is equivalent to qgr of the reconstruction algebra, and that the degree zero piece of the reconstruction algebra coincides with Ringel's canonical algebra. This implies that the reconstruction algebra contains the canonical algebra, and furthermore its qgr category is derived equivalent to the canonical algebra, thus linking the reconstruction algebra of rational surface singularities to the canonical algebra of representation theory.
| Item Type: | Articles |
|---|---|
| Status: | Published |
| Refereed: | Yes |
| Glasgow Author(s) Enlighten ID: | Wemyss, Professor Michael |
| Authors: | Iyama, O., and Wemyss, M. |
| College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |
| Journal Name: | Épijournal de Géométrie Algébrique |
| Publisher: | Association Epiga |
| ISSN: | 2491-6765 |
| ISSN (Online): | 2491-6765 |
| Copyright Holders: | Copyright © 2018 The Authors |
| First Published: | First published in Épijournal de Géométrie Algébrique 3:13 |
| Publisher Policy: | Reproduced under a Creative Commons licence |
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