A criterion for virtual global generation

Abstract

Let X be a smooth projective curve defined over an algebraically closed field k, and let FX denote the absolute Frobenius morphism of X when the characteristic of k is positive. A vector bundle over X is called virtually glob- ally 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 ∼ Ea ⊕ E f for some m, where Ea is some ample vector bundle and E f is some finite vector bundle over X (either of Ea 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).