Affine semipartial geometries and projections of quadrics

Debroey and Thas introduced semipartial geometries and determined the full embeddings of semipartial geometries in AG(n,q) for n=2 and 3. For n>3 there is no such classification. A model of a semipartial geometry fully embedded in AG(4,q), q even, due to Hirschfeld and Thas, is the spg(q−1,q2,2,2q(q−1)) constructed by projecting the quadric Q−(5,q) from a point of PG(5,q)⧹Q−(5,q). In this paper this semipartial geometry is characterized amongst the spg(q−1,q2,2,2q(q−1)) (of which there is an infinite family of non-classical examples due to Brown) by its full embedding in AG(4,q).