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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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