# A criterion for virtual global generation

Indranil Biswas; A. J. Parameswaran

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

- Volume: 5, Issue: 1, page 39-53
- ISSN: 0391-173X

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topBiswas, Indranil, and Parameswaran, A. J.. "A criterion for virtual global generation." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.1 (2006): 39-53. <http://eudml.org/doc/240363>.

@article{Biswas2006,

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 $(F^m_X)^* E\, \cong \, E_a\oplus E_f$ 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).},

author = {Biswas, Indranil, Parameswaran, A. J.},

journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},

language = {eng},

number = {1},

pages = {39-53},

publisher = {Scuola Normale Superiore, Pisa},

title = {A criterion for virtual global generation},

url = {http://eudml.org/doc/240363},

volume = {5},

year = {2006},

}

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

PB - Scuola Normale Superiore, Pisa

VL - 5

IS - 1

SP - 39

EP - 53

AB - 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 $(F^m_X)^* E\, \cong \, E_a\oplus E_f$ 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).

LA - eng

UR - http://eudml.org/doc/240363

ER -

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