The classification of reflexive modules of rank one over rational and minimally elliptic singularities

TL;DR

This paper classifies reflexive modules of rank one over rational and minimally elliptic singularities, extending classical correspondences and confirming conjectures about flat connections in cusp singularities.

Contribution

It provides a comprehensive classification of rank one reflexive modules over these singularities, including special and flat modules, and generalizes existing classical results.

Findings

Classified all rank one reflexive modules over rational and minimally elliptic singularities.

Confirmed that all reflexive modules over cusp singularities admit flat connections.

Extended classical McKay correspondence and related results to broader singularity classes.

Abstract

We classify the reflexive modules of rank one over rational and minimally elliptic singularities. Equivalently, we classify full line bundles on the resolutions of rational and minimally elliptic singularities. As an application, we determine among such reflexive modules of rank one all the special ones (in the sense of Wunram) and all the flat ones. In this way, we also classify the non-flat reflexive modules as well (as a generalization of a construction of Dan and Romano). In particular, we prove (in the rank one case) a conjecture of Behnke, namely that in the case of a cusp singularity any reflexive module admits a flat connection. The results generalize the classical Mckay correspondence, and results of Artin, Verdier, Esnault, Khan and Wunram valid for different particular families of singularities.