EMBEDDINGS OF POINT-LINE GEOMETRIES

Embeddings. Let be a point-line geometry, that is, an incidence system of points and lines, such that distinct lines possess distinct point-shadows, thus allowing lines to be viewed as sets of points. A projective embedding of point-line geometry Γ into the projective space P(V) of all proper subspaces of the vector space V is an injective mapping → projective points of P(V) = 1-spaces of V such that

1. (1) e(L) is a projective line for each line L of, and

2. (2) the image points span P(V).

Such an embedding is denoted by the symbol e : Γ → P(V).

Morphisms of embeddings. Let τ : V → W be a semilinear transformation of vector spaces. This induces a partial mapping of the corresponding projective spaces P(V) and P(W), sending points of P(V) not contained in kerr in P(V) to projective points of P(W). With some abuse of notation, we denote this by τ : P(V) → P(W). If e : Γ → P(V) is a projective embedding of the point-line geometry Γ, then composition with the partial map τ can yield a new embedding eτ if and only if

1. (1) τ is a surjective semilinear transformation, and

2. (2) For any points p and q of Γ, ker τ meets the subspace < e(p), e(q) > at the zero subspace of V.