Geodesic
Triviality Properties of Principal Bundles on Singular Curves. II
Canadian mathematical bulletin, Tome 63 (2020) no. 2, pp. 423-433

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DOI

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 $S$.
DOI : 10.4153/S0008439519000523
Mots-clés : principal bundle, singular curve, obstructions to triviality 
Belkale, P.; Fakhruddin, N. Triviality Properties of Principal Bundles on Singular Curves. II. Canadian mathematical bulletin, Tome 63 (2020) no. 2, pp. 423-433. doi: 10.4153/S0008439519000523
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     author = {Belkale, P. and Fakhruddin, N.},
     title = {Triviality {Properties} of {Principal} {Bundles} on {Singular} {Curves.} {II}},
     journal = {Canadian mathematical bulletin},
     pages = {423--433},
     year = {2020},
     volume = {63},
     number = {2},
     doi = {10.4153/S0008439519000523},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000523/}
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