Weighted Projective Lines and Rational Surface Singularities
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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Association Epiga
2020-01-01
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| Series: | Épijournal de Géométrie Algébrique |
| Subjects: | |
| Online Access: | https://epiga.episciences.org/4761/pdf |
| _version_ | 1811238187904794624 |
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| author | Osamu Iyama Michael Wemyss |
| author_facet | Osamu Iyama Michael Wemyss |
| author_sort | Osamu Iyama |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-12T12:37:32Z |
| format | Article |
| id | doaj.art-9540d18b2a334e2a813718fdbdac558a |
| institution | Directory Open Access Journal |
| issn | 2491-6765 |
| language | English |
| last_indexed | 2024-04-12T12:37:32Z |
| publishDate | 2020-01-01 |
| publisher | Association Epiga |
| record_format | Article |
| series | Épijournal de Géométrie Algébrique |
| spelling | doaj.art-9540d18b2a334e2a813718fdbdac558a2022-12-22T03:32:51ZengAssociation EpigaÉpijournal de Géométrie Algébrique2491-67652020-01-01Volume 310.46298/epiga.2020.volume3.47614761Weighted Projective Lines and Rational Surface SingularitiesOsamu IyamaMichael WemyssIn 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.https://epiga.episciences.org/4761/pdfmathematics - representation theorymathematics - commutative algebramathematics - algebraic geometry |
| spellingShingle | Osamu Iyama Michael Wemyss Weighted Projective Lines and Rational Surface Singularities Épijournal de Géométrie Algébrique mathematics - representation theory mathematics - commutative algebra mathematics - algebraic geometry |
| title | Weighted Projective Lines and Rational Surface Singularities |
| title_full | Weighted Projective Lines and Rational Surface Singularities |
| title_fullStr | Weighted Projective Lines and Rational Surface Singularities |
| title_full_unstemmed | Weighted Projective Lines and Rational Surface Singularities |
| title_short | Weighted Projective Lines and Rational Surface Singularities |
| title_sort | weighted projective lines and rational surface singularities |
| topic | mathematics - representation theory mathematics - commutative algebra mathematics - algebraic geometry |
| url | https://epiga.episciences.org/4761/pdf |
| work_keys_str_mv | AT osamuiyama weightedprojectivelinesandrationalsurfacesingularities AT michaelwemyss weightedprojectivelinesandrationalsurfacesingularities |