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Triviality Properties of Principal Bundles on Singular Curves. II

Part of: Families, fibrations Curves

Published online by Cambridge University Press:  24 January 2020

P. Belkale  and
N. Fakhruddin
P. Belkale
Affiliation:
Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA Email: belkale@email.unc.edu
N. Fakhruddin
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India Email: naf@math.tifr.res.in
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Abstract

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\rightarrow S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D\subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected, but it can be made to vanish by a faithfully flat base change, if $S$ is the spectrum of a dvr (and some other hypotheses). The vanishing of this obstruction is shown to be a sufficient condition for étale local triviality if $S$ is a smooth curve, and the singular locus of $X-D$ is finite over $S$.

Keywords

principal bundlesingular curveobstructions to triviality

MSC classification

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles
Secondary: 14H60: Vector bundles on curves and their moduli 14H20: Singularities, local rings
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 2 , June 2020 , pp. 423 - 433
Copyright
© Canadian Mathematical Society 2019

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