Embeddings of Cotriangular Spaces

Abstract

AbstractA cotriangular space is a partial linear space with three points per line, such that a point is collinear to 0 or 2 points on the line. Such a space is called irreducible if it is connected, and no two points are collinear with the same set of points. Results by Hall and Shult imply that irreducible cotriangular spaces can be subdivided into three classes: those of triangular type, those of symplectic type, and those of orthogonal type. In these spaces any two intersecting lines are in a dual affine plane. We consider embeddings of cotriangular spaces in which any two intersecting lines are in a dual affine plane into projective spaces over arbitrary fields. We show that if such a cotriangular space admits an embedding over a field \({\mathbb F}\), it admits a universal embedding over the field \({\mathbb F}\). Moreover we classify all universal embeddings of irreducible cotriangular spaces. For the spaces of symplectic or orthogonal type we describe, if the characteristic \({\mathbb F}\) is two, the embedding using the associated quadratic forms. For other characteristics the universal embeddings only exist for cotriangular spaces associated to the root systems of type \(E_{6}\), \(E_{7}\), and \(E_{8}\) and to root systems of type \(A_{n}\), where \(n>4\) or \(n=\infty \).KeywordsEmbeddingCotriangular spacesRoot lattices