×

Triviality properties of principal bundles on singular curves. II. (English) Zbl 1454.14093

Summary: 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 \(S\).
For part I, see [P. Belkale and N. Fakhruddin, Algebr. Geom. 6, No. 2, 234–259 (2019; Zbl 1444.14036)].

MSC:

14H60 Vector bundles on curves and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
14H20 Singularities of curves, local rings

Citations:

Zbl 1444.14036
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Barth, W. P., Hulek, K., Peters, C. A. M., and Van De Ven, A., Compact complex surfaces, Second ed., , Springer-Verlag, Berlin, 2004. · Zbl 1036.14016
[2] Beauville, A. and Laszlo, Y., Un lemme de descente. C. R. Acad. Sci. Paris Sér. I Math.320(1995), no. 3, 335-340. · Zbl 0852.13005
[3] Belkale, P. and Fakhruddin, N., Triviality properties of principal bundles on singular curves. Algebr. Geom.6(2019), 234-259. · Zbl 1444.14036
[4] Deligne, P., Milne, J. S., Ogus, A., and Shih, K.-Y., Hodge cycles, motives, and Shimura varieties. , Springer, Berlin-New York, 1982. · Zbl 0465.00010
[5] Drinfeld, V. G. and Simpson, C., B-structures on G-bundles and local triviality. Math. Res. Lett.2(1995), no. 6, 823-829. · Zbl 0874.14043
[6] Faltings, G., A proof for the Verlinde formula. J. Algebraic Geom.3(1994), 347-374. · Zbl 0809.14009
[7] Grothendieck, A., Le groupe de Brauer. II. Théorie cohomologique. In: Dix Exposés sur la Cohomologie des Schémas. , North-Holland, Amsterdam; Masson, Paris, 1968, pp. 67-87. · Zbl 0192.57801
[8] Hartshorne, R., Algebraic geometry. , Springer, New York-Heidelberg, 1977. · Zbl 0367.14001
[9] Lipman, J., Desingularization of two-dimensional schemes. Ann. Math. (2)107(1978), 151-207. · Zbl 0349.14004
[10] Solis, P., A wonderful embedding of the loop group. Adv. Math.313(2017), 689-717. · Zbl 1367.14018
[11] Solis, P., Nodal uniformization of G-bundles. 2016. arxiv:1608.05681
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.