模式值点的交比
Cross-Ratios of Scheme-Valued Points
摘要：
交比的经典理论是研究射影直线有序点的模和$PGL_2$作用的不变量的一个很好的例子。本文将交比理论推广到任意格式$S$的$S$ -值点的设置。为了实现这一目标，我们提供了一个全面的和计算集中的治疗的自同构的射影空间在$S$，均衡器的类别的计划，和消失轨迹的部分线丛。这些想法大多存在于文献中，尽管没有达到我们所要求的详细程度或普遍性。在介绍了“强不同”的概念对态射，我们定义的交比4元组的成对强不同的$S$值点的投影线-这是价值的单位环的整体功能的计划$S$ -并表明，它享有所有熟悉的性质的交比。
英文摘要：
The classical theory of the cross-ratio is a beautiful case study of the moduli of ordered points of the projective line and of invariants of the action of $PGL_2$. We generalize the theory of the cross-ratio to the setting of $S$-valued points for an arbitrary scheme $S$. To accomplish this goal, we provide a comprehensive and computationally focused treatment of automorphisms of projective space over $S$, of equalizers in the category of schemes, and of vanishing loci of sections of line bundles. Most of these ideas exist in the literature, though not with the level of detail or generality that we require. After introducing the notion of a "strongly distinct" pair of morphisms, we define the cross-ratio of 4-tuples of pairwise strongly distinct $S$-valued points of the projective line -- which is valued in the units of the ring of global functions on the scheme $S$ -- and show that it enjoys all of the familiar properties of the cross-ratio.
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