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A criterion for virtual global generation. (English) Zbl 1170.14308
Summary: 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).

##### MSC:
 14H60 Vector bundles on curves and their moduli 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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