We describe three algorithms to determine the stable, semistable, and torus-polystable loci of the GIT quotient of a projective variety by a reductive group. The algorithms are efficient when the group is semisimple. By using an implementation of our algorithms for simple groups, we provide several applications to the moduli theory of algebraic varieties, including the K-moduli of algebraic varieties, the moduli of algebraic curves and the Mukai models of the moduli space of curves for low genus. We also discuss a number of potential improvements and some natural open problems arising from this work.

翻译：我们描述了三种算法，用于确定射影簇在约化群作用下GIT商中的稳定点、半稳定点及环面-多稳定点的位置。当群为半单群时，这些算法具有较高效率。通过实现针对单群的算法，我们为代数簇模空间理论提供了若干应用，包括代数簇的K-模空间、代数曲线模空间以及低亏格曲线模空间的Mukai模型。此外，本文还讨论了若干潜在的改进方向以及由本研究自然引出的一些开放性问题。