Degeneration of torsors over families of del Pezzo surfaces

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 generi...

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Bibliographic Details
Published in:Mathematische Nachrichten 2020-12, Vol.293 (12), p.2306-2334
Main Authors: Derenthal, Ulrich, Hoffmann, Norbert
Format: Article
Language:English
Subjects:
Degeneration
del Pezzo surface
Singularity (mathematics)
universal torsor
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Summary: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.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201900546