相对几何不变量理论：约化与非约化
Relative Geometric Invariant Theory: Reductive and Non-reductive
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中文总结 AI 辅助
本文通过构造良好商并给出希尔伯特-芒福德描述，统一处理约化与非约化群作用的几何不变量理论，并应用于不稳定对象模空间、带重数箭图表示及射流。
中文摘要 AI 辅助
我们为群同态在概形态射上的等变作用构造了良好商。利用几何不变量理论，我们在源中获得了显式的开半稳定轨迹，并带有希尔伯特-芒福德描述，这些轨迹相对于目标的一个给定良好商承认良好商。特别地，我们获得了约化群作用于仿射上射影态射的商。在非约化情形下，我们考虑仿射态射上的等变作用，这些作用由乘法群‘分级’并满足某些幺幂稳定子假设。这作为特例恢复了射影非约化几何不变量理论中的已知结果，同时也证明了该设定下的希尔伯特-芒福德判据。作为应用，我们考虑了不稳定对象的模空间、带重数的箭图表示以及射流。
英文摘要
We construct good quotients for equivariant actions of group homomorphisms on morphisms of schemes. Using Geometric Invariant Theory, we obtain explicit open semistable loci in the source with Hilbert-Mumford descriptions admitting good quotients relative to a given good quotient of the target. In particular, we obtain quotients for reductive groups acting on projective-over-affine morphism. In the non-reductive case, we consider equivariant actions on affine morphisms which are 'graded' by a multiplicative group and satisfy certain unipotent stabiliser assumptions. This recovers known results in projective non-reductive GIT as a special case, which also proves the Hilbert-Mumford criterion in that setting. As applications, we consider moduli of unstable objects, representations of quivers with multiplicities and jets.
