加权射影直线与有理曲面奇点
Weighted Projective Lines and Rational Surface Singularities
AI总结：
本文在温和假设下，通过加权射影直线上的Veronese子环显式描述有理曲面奇点的特殊Cohen-Macaulay模，并证明其重构代数与典范代数的联系。
摘要：
本文研究了具有星形对偶图的有理曲面奇点R，并在对自交数非常温和的假设下，给出了其所有特殊Cohen-Macaulay模的显式描述。我们通过将R实现为Geigle-Lenzing加权射影直线X的齐次坐标环S的某个Z-分次Veronese子环S^x来实现这一点，并将特殊CM模实现为X上典范倾斜丛的显式描述的直和项。然后，我们通过比较qgr S^x和coh X给出了这些是特殊CM模的第二个证明，并且我们还给出了这两个范畴等价的充分必要条件组合判据。反过来，我们证明qgr S^x等价于重构代数的qgr，并且重构代数的零次部分与Ringel的典范代数重合。这意味着重构代数包含典范代数，此外，其qgr范畴在导出意义下等价于典范代数，从而将有理曲面奇点的重构代数与表示论中的典范代数联系起来。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
