A criterion for virtual global generation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 1, p. 39-53

Let X be a smooth projective curve defined over an algebraically closed field k, and let F X denote the absolute Frobenius morphism of X when the characteristic of k is positive. A vector bundle over X is called virtually globally generated if its pull back, by some finite morphism to X from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of k is positive, a vector bundle E over X is virtually globally generated if and only if (F X m ) * EE a E f for some m, where E a is some ample vector bundle and E f is some finite vector bundle over X (either of E a and E f are allowed to be zero). If the characteristic of k is zero, a vector bundle E over X is virtually globally generated if and only if E is a direct sum of an ample vector bundle and a finite vector bundle over X (either of them are allowed to be zero).

Classification:  14H60,  14F05
@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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {1},
     year = {2006},
     pages = {39-53},
     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, Serie 5, Volume 5 (2006) no. 1, pp. 39-53. http://www.numdam.org/item/ASNSP_2006_5_5_1_39_0/

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