On the local recognition of finite metasymplectic spaces (Q1823468)

A spherical building \(\Omega\) of type \(M_ n\) (rank n) is up to isomorphism uniquely determined by its shadow space \(\Delta\) with respect to any node r of M (points of \(\Delta\) are the vertices of \(\Omega\) of type r and the lines of \(\Delta\) are the shadows of the chambers). For every point p of \(\Delta\) there is a subgeometry \(\Delta_{\leq 1}(p)\) which is the subspace containing all points collinear with p. For different p, they are all isomorphic. One calls \(\Delta\) locally recognizable if it is completely determined by \(\Delta_{\leq 1}(p)\) in the usual sense. The paper shows two main things: (1) Certain shadow spaces of buildings of type \(D_ n\) (4\(\leq n\leq 7)\); \(E_ 6\), \(E_ 7\), \(E_ 8\) are local recognizable. (2) The main result however is that the shadow space of buildings of type \(F_ 4\) with respect to an end node (the so called metasymplectic spaces) are local recognizable under a certain condition (made precise in the paper) which is trivially satisfied in the thin case (thus extending the result of \textit{D. Buset}, Discrete Math. 46, 221-226 (1983; Zbl 0532.05050)). Along the proof, the authors also give a characterization of locally quad subspaces of a dual polar space of rank 3.