Cross-Ratios of Scheme-Valued Points

TL;DR

This paper extends the classical cross-ratio concept to scheme-valued points, providing detailed theoretical foundations and defining a generalized cross-ratio with properties analogous to the classical case.

Contribution

It introduces a comprehensive framework for defining and analyzing the cross-ratio of scheme-valued points, generalizing classical invariants to arbitrary schemes.

Findings

Defined the cross-ratio for scheme-valued points with classical properties

Developed detailed methods for automorphisms and equalizers over schemes

Established the cross-ratio as a scheme-invariant with algebraic properties

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.