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}, zbl = {1170.14308}, mrnumber = {2240182}, language = {en}, url = {http://www.numdam.org/item/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, Série 5, Tome 5 (2006) no. 1, pp. 39-53. http://www.numdam.org/item/ASNSP_2006_5_5_1_39_0/
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