AI 中文总结
该研究探讨弗罗贝尼乌斯迹核丰沛的光滑射影簇，证明其非常态态射必有限且应为皮卡秩1的法诺簇，还证实经典型和G₂型广义格拉斯曼簇的弗罗贝尼乌斯迹核丰沛，仅低特征例外。
AI 中文摘要
我们研究光滑射影簇，其弗罗贝尼乌斯迹核（又称与弗罗贝尼乌斯态射相关的切恩豪森丛）是丰沛的。我们证明，这类簇的任何非常态态射都必须在其像上是有限的，为这些簇必定是皮卡秩1的法诺簇提供了有力证据。利用无穷小表示理论并通过构造特殊的弗罗贝尼乌斯分裂，我们证明经典型和G₂型的广义格拉斯曼簇具有丰沛的弗罗贝尼乌斯迹核，仅存在某些低特征例子例外，这些例外由相关代数群的外同构存在性所解释。
英文摘要
We study smooth projective varieties whose Frobenius-trace kernel, also known as the Tschirnhausen bundle associated with the Frobenius morphism, is ample. We show that any non-constant morphism out of such a variety must be finite onto its image, providing strong evidence that these varieties must be Fano varieties of Picard rank 1. Using infinitesimal representation theory and by constructing special Frobenius splittings, we show that generalized Grassmannian of classical type and type $\mathrm{G}_2$ have ample Frobenius-trace kernel, with the exception of certain low characteristic examples which are explained by the existence of exotic isogenies of the associated algebraic groups.
Comments28 pages, comments very welcome!
