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.

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 ${\left({F}_{X}^{m}\right)}^{*}E\phantom{\rule{0.166667em}{0ex}}\cong \phantom{\rule{0.166667em}{0ex}}{E}_{a}\oplus {E}_{f}$ 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},
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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