Related papers: Cross-Ratios of Scheme-Valued Points
In one complex variable, the cross ratio is a well-known quantity associated with four given points in the complex plane that remains invariant under linear fractional maps. In particular, if one knows where three points in the complex…
We define a family of four-point invariants for Shilov boundaries of bounded symmetric domains of tube type, which generalizes the classical four-point cross ratio on the unit circle. This generalization, which is based on a similar…
We develop a complete theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra (PGA), R(n,0,1), covering geometric objects of every grade: finite and ideal points, hyperplanes, and intermediate flats. For each object…
Cross-ratio degrees count configurations of points $z_1,\ldots, z_n \in \mathbb{P}^1$ satisfying $n - 3$ cross-ratio constraints, up to isomorphism. These numbers arise in multiple contexts in algebraic and tropical geometry, and may be…
For a certain class of configurations of points in space, Eves' Theorem gives a ratio of products of distances that is invariant under projective transformations, generalizing the cross-ratio for four points on a line. We give a…
A compact classification of the projective lines defined over (commutative) rings (with unity) of all orders up to thirty-one is given. There are altogether sixty-five different types of them. For each type we introduce the total number of…
The space of n (ordered) points on the projective line, modulo automorphisms of the line, is one of the most important and classical examples of an invariant theory quotient, and is one of the first examples given in any course. Generators…
We investigate the enumerative geometry of point configurations in projective space. We define "projective configuration counts": these enumerate configurations of points in projective space such that certain specified subsets are in fixed…
The paper presents a new cross-ratio of hypercomplex numbers based on projective geometry. We discuss the essential properties of the projective cross-ratio, notably its invariance under Mobius transformations. Applications to the geometry…
The main result is a wall crossing formula for central projections defined on submanifolds of a real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the…
A generalization of von Staudt's theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the…
Motivated by a M\"obius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular M\"obius invariant…
We consider the ring of invariants of n points on the projective line. The space (P^1)^n // PGL_2 is perhaps the first nontrivial example of a Geometry Invariant Theory quotient. The construction depends on the weighting of the n points.…
When considering geometry, one might think of working with lines and circles on a flat plane as in Euclidean geometry. However, doing geometry in other spaces is possible, as the existence of spherical and hyperbolic geometry demonstrates.…
In this article we generalize a classical result by Passman on primeness of unital strongly group graded rings to the class of nearly epsilon-strongly group graded rings which are not necessarily unital. Using this result, we obtain (i) a…
Our approach to structural matrix rings defines them over preordered directed graphs. A grading of a structural matrix ring is called a good grading if its standard unit matrices are homogeneous. For a group $G$, a $G$ -grading set is a set…
We prove a theorem on distortion of cross ratio of four points under the mapping effected by a complex polynomial with restricted critical values. Its corollaries include inequalities involving the absolute value and certain coefficients of…
The paper introduces cycles cross ratio, which extends the classic cross ratio of four points to various settings: conformal geometry, Lie spheres geometry, etc. Just like its classic counterpart cycles cross ratio is a measure of…
First we give a complex ball uniformization of the moduli space of 8 ordered points on the projective line by using the theory of periods of K3 surfaces. Next we give a projective model of this moduli space by using automorphic forms on a…
This paper proves that the characteristic polynomial is a complete unitary invariant for pairs of projection matrices. Some special cases involving three or more projections are also considered.