Virtual global generation in higher dimensions
Indranil Biswas, Manish Kumar, A. J. Parameswaran
TL;DR
The paper investigates how the notion of virtual global generation (VGG) for vector bundles on curves extends to higher-dimensional, normal projective varieties. It introduces and analyzes several variants—$m$-VGG, strongly VGG, curve VGG, and the VGG condition via global generation of $\mathcal{O}_{\mathbb{P}(E)}(N)$—establishing implications such as strongly VGG $\Rightarrow$ VGG $\Rightarrow$ curve VGG, and showing that on curves these conditions are all equivalent. A precise treatment of line bundles reveals that strongly VGG is equivalent to global generation of a suitable tensor power, with a proof leveraging Frobenius and Galois covers. The paper provides explicit higher-dimensional examples showing that curve VGG does not imply VGG, and constructs $m$-VGG line bundles that fail to be $(m+1)$-VGG, thereby clarifying the landscape of VGG generalizations and their limitations.
Abstract
The notion of virtual global generation (VGG) for a vector bundle has multiple possible generalization from the case of curves to higher dimensional normal projective varieties. We study relationship between these notions. All these notions agree for curves but in higher dimension we show that this is not the case.