A criterion for virtual global generation
A criterion for virtual global generation
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虚拟全局生成的标准
DOI：
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发表时间：
2006
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影响因子：
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通讯作者：
A. J. Parameswaran
中科院分区：
文献类型：
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作者：
I. Biswas;A. J. Parameswaran
设X是定义在代数闭域k上的光滑射影曲线，FX表示X在特征线k为正时的绝对Frobenius态射. X上的向量丛称为虚全局生成的，如果它的拉回，通过从某个光滑射影曲线到X的某个有限态射，由它的全局截面生成。我们证明了以下几点。如果k的特征是正的，则X上的向量丛E实际上是全局生成的当且仅当（F m X）<$E <$Ea <$E f对于某个m，其中Ea是某个充足向量丛，E f是X上的某个有限向量丛（Ea和E f中的任何一个都允许为零）。如果k的特征为零，则X上的向量丛E是虚拟全局生成的，当且仅当E是X上的充足向量丛和有限向量丛的直和（允许它们中的任何一个为零）。
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).