On subregular slices of the elliptic Grothendieck–Springer resolution

We study singularities, resolutions and deformations coming from subregular slices of the elliptic Grothendieck–Springer resolution constructed by the author in [5] (The elliptic Grothendieck–Springer resolution as a simultaneous log resolution of algebraic stacks, 2019. arXiv 1908.04140). This is a simultaneous log resolution of an extended coarse moduli space map with domain the stack of principal bundles on an elliptic curve with simply connected simple structure group. We construct explicit slices of this stack through all subregular unstable bundles, for all possible structure groups. When the structure group is not $SL_2$, we describe the pullbacks of the elliptic Grothendieck–Springer resolution to these slices as concrete varieties, extending and refining earlier work of I. Grojnowski and N. Shepherd–Barron, who related these varieties for exceptional structure groups to del Pezzo surfaces. We use the resolutions to identify the singularities of the unstable locus of the subregular slices, and prove that the extended coarse moduli space map gives torus-equivariant deformations that are miniversal among those with appropriately restricted weights.

Keywords

singularities, principal bundles, elliptic curves

2010 Mathematics Subject Classification

Primary 14H60. Secondary 14J17, 14L35, 14L40.

Full Text (PDF format)

The author was supported by King’s College London, and the EPSRC grants [EP/L015234/1] (The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London) and [EP/R034826/1].

Received 13 August 2020

Accepted 24 September 2021

Published 26 January 2022