论文标题
方案值点的交叉比例
Cross-Ratios of Scheme-Valued Points
论文作者
论文摘要
交叉比例的经典理论是对投影线的有序点和$ pgl_2 $行动的有序点的模量的精美案例研究。我们将交叉比例的理论推广到任意方案$ s $的$ S $值点的设置。为了实现这一目标，我们提供了针对$ s $ $ s $，方案类别的均衡器的投射空间自动形态的全面和计算重点的处理，并消失了线条捆绑包的基因座。这些想法中的大多数都存在于文献中，尽管不是我们需要的细节或一般性水平。在介绍了“强烈不同”的形态的概念之后，我们定义了投影线的4个成对$ s $可值点的4个tublatio的交叉比例 - 这是在该方案$ s $上的全球功能单位中的价值，并表明它享受着熟悉熟悉的交叉会计属性。
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.
