Let be a smooth projective curve defined over an algebraically closed field , and let denote the absolute Frobenius morphism of when the characteristic of is positive. A vector bundle over is called virtually globally generated if its pull back, by some finite morphism to from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of is positive, a vector bundle over is virtually globally generated if and only if for some , where is some ample vector bundle and is some finite vector bundle over (either of and are allowed to be zero). If the characteristic of is zero, a vector bundle over is virtually globally generated if and only if is a direct sum of an ample vector bundle and a finite vector bundle over (either of them are allowed to be zero).
@article{ASNSP_2006_5_5_1_39_0, author = {Biswas, Indranil and Parameswaran, A. J.}, title = {A criterion for virtual global generation}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {39--53}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 5}, number = {1}, year = {2006}, mrnumber = {2240182}, zbl = {1170.14308}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2006_5_5_1_39_0/} }
TY - JOUR AU - Biswas, Indranil AU - Parameswaran, A. J. TI - A criterion for virtual global generation JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2006 SP - 39 EP - 53 VL - 5 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2006_5_5_1_39_0/ LA - en ID - ASNSP_2006_5_5_1_39_0 ER -
%0 Journal Article %A Biswas, Indranil %A Parameswaran, A. J. %T A criterion for virtual global generation %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2006 %P 39-53 %V 5 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2006_5_5_1_39_0/ %G en %F ASNSP_2006_5_5_1_39_0
Biswas, Indranil; Parameswaran, A. J. A criterion for virtual global generation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 1, pp. 39-53. http://archive.numdam.org/item/ASNSP_2006_5_5_1_39_0/
[Ba] Tensor products of ample vector bundles in characteristic , Amer. J. Math. 93 (1971), 429-438. | MR | Zbl
,[Bi1] Parabolic ample bundles, Math. Ann. 307 (1997), 511-529. | MR | Zbl
,[Bi2] A criterion for ample vector bundles over a curve in positive characteristic, Bull. Sci. Math. 129 (2005), 539-543. | MR | Zbl
,[BP] On the ample vector bundles over curves in positive characteristic, C.R. Acad. Sci. Paris Sér. I Math. 339 (2004), 355-358. | MR | Zbl
and ,[Br] Slopes of vector bundles on projective curves and applications to tight closure problems, Trans. Amer. Math. Soc. 356 (2004), 371-392. | MR | Zbl
,[Ha1] “Ample Subvarieties of Algebraic Varieties”, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. | MR | Zbl
,[Ha2] Ample vector bundles on curves, Nagoya Math. J. 43 (1971), 73-89. | MR | Zbl
,[La] On stable and ample vector bundles of rank 2 on curves, Math. Ann. 238 (1978), 193-202. | MR | Zbl
,[Lan] Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), 251-276. | MR | Zbl
,[MS] Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), 205-239. | MR | Zbl
and ,[Mu] Pathologies III, Amer. J. Math. 89 (1967), 94-104. | MR | Zbl
,[No] On the representations of the fundamental group scheme, Compositio Math. 33 (1976), 29-41. | Numdam | MR | Zbl
,[Pa] Virtual global generation of ample bundles over curves, unpublished manuscript.
,[RR] Some remarks on the instability flag, Tôhoku Math. J. (2) 36 (1984), 269-291. | MR | Zbl
and ,