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Triviality Properties of Principal Bundles on Singular Curves. II
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 63 / Issue 2 / June 2020
- Published online by Cambridge University Press:
- 24 January 2020, pp. 423-433
- Print publication:
- June 2020
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- Article
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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 
$S$.
