AI 中文总结
本文为仿射超概形在约化代数超群作用下发展几何不变量理论，构造仿射超商，证明超几何希尔伯特-芒福德判据，并展示其用于构造GIT超商及与经典情形的差异。
AI 中文摘要
我们发展了在约化代数超群作用下的仿射超概形的几何不变量理论，通过坐标Hopf超代数来表述该理论，以适应反交换变量和幂零变量。在此代数框架内，我们构造了仿射超商，并在适当假设下确立了其基本结构性质。随后，我们证明了希尔伯特-芒福德判据的超几何类比，在超群余作用的自然约束下，给出了点的半稳定性和稳定性的数值检验。我们表明，该数值判据可用于构造这些超概形的GIT超商，并举例说明超几何GIT超商与其经典对应物的差异。
英文摘要
We develop Geometric Invariant Theory for affine superschemes under the action of reductive algebraic supergroups, formulating the theory in terms of coordinate Hopf superalgebras in order to accommodate anticommuting and nilpotent variables. Within this algebraic setting, we construct affine superquotients and establish their basic structural properties under suitable hypotheses. We then prove a supergeometric analogue of the Hilbert-Mumford criterion, giving a numerical test for the semistability and stability of points under natural constraints on the supergroup coaction. We show that this numerical criterion can be used to construct the GIT superquotient of these superschemes, and we give examples illustrating how the supergeometric GIT superquotient differs from its classical counterpart.
Comments43 pages, comments and questions are very welcome
