Weighted Projective Lines and Rational Surface Singularities

TL;DR

This paper explores the structure of rational surface singularities with star-shaped dual graphs by linking them to weighted projective lines, providing explicit descriptions of special Cohen-Macaulay modules and connecting their algebraic properties to canonical algebras.

Contribution

It introduces a novel approach to describe special Cohen-Macaulay modules via weighted projective lines and establishes categorical equivalences with reconstruction and canonical algebras.

Findings

Explicit description of special Cohen-Macaulay modules for certain surface singularities.

Equivalence between categories of graded rings and coherent sheaves on weighted projective lines.

Reconstruction algebra contains the canonical algebra and is derived equivalent to it.

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.