On subregular slices of the elliptic Grothendieck-Springer resolution (Q2071467)

The additive Grothendieck-Springer resolution (see diagram (1.0.2)) serves as a resolution of singularities in the study of du Val singularities for algebraic surfaces. Given the stack of principal \(G\)-bundles over an elliptic curve with a simply connected structure group \(G\), this article describes the singularities and log resolutions coming from the elliptic Grothendieck-Springer resolution (see diagram (1.0.3)), which is a stacky version of the additive one. When \(G\neq SL_2\), Theorem 1.0.2 (proven in section 2) shows the existence of an equivariant slice through subregular unstable bundles with good properties. Theorems 1.0.3 (proven in section 3) and 1.0.6 (proven in section 4) gives explicit descriptions of the pullbacks to these slices, depending on the Dynkin type of the subregular unstable \(G\)-bundle, and the singular surfaces appearing on the fibrations. This paper contains the results of chapters 5 and 6 of the author's PhD thesis.