Abstract:Recognizing specific geometric structures by certain properties - prefer- ably as weak as possible - is very important in finite geometry, since various configurations can turn up in completely different contexts. Data that are available for structures in projective spaces are in many situations the inter- section numbers with respect to subspaces. Our aim is to characterize the standard embeddings of the split Cayley hexagon H (q ) in PG(5; q ), with q even, and in PG(6; q ), by intersection numbers. Since in these representations the points of H (q ) are all the points of PG(5; q ), respectively all the points of a parabolic quadric of PG(6; q ), such a characterization is impossible if we only consider intersections of the point set of the hexagon with subspaces. That is why we consider intersections of subspaces with the line set of H (q ). We obtain very strong results, several of them brand new. Also, a new type of geometry turned up, and these geometries will be called Singer geometries.