Cross-Ratios of Scheme-Valued Points

Abstract: 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.