Embedding a Latin square with transversal into a projective space

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n 2 lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3 n + 1 points, with three lines of size n, n 2 − n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n 2 lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser. A 92 (2000) 88–94], we characterise embeddings of these finite geometries into projective spaces over skew fields.