A criterion for virtual global generation

Biswas, Indranil; Parameswaran, A. J.

Biswas, Indranil; Parameswaran, A. J.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 1, p. 39-53

Abstract

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 ${(F_{X}^{m})}^{*} E ≅ E_{a} \oplus E_{f}$ 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).

MR 2240182 | Zbl 1170.14308

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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