Degeneration of torsors over families of del Pezzo surfaces

TL;DR

This paper constructs a G-torsor over a family of del Pezzo surfaces with ADE singularities, extending previous work and establishing uniqueness and rigidity in good residue characteristics.

Contribution

It extends the construction of G-torsors from individual singular del Pezzo surfaces to families over discrete valuation rings, ensuring existence, uniqueness, and rigidity.

Findings

Constructed a G-torsor over the family S.

Proved the torsor extends the universal torsor on the generic fiber.

Established uniqueness and infinitesimal rigidity in very good residue characteristic.

Abstract

Let S be a split family of del Pezzo surfaces over a discrete valuation ring such that the general fiber is smooth and the special fiber has ADE-singularities. Let G be the reductive group given by the root system of these singularities. We construct a G-torsor over S whose restriction to the generic fiber is the extension of structure group of the universal torsor. This extends a construction of Friedman and Morgan for individual singular del Pezzo surfaces. In case of very good residue characteristic, this torsor is unique and infinitesimally rigid.