Semipartial geometries, arising from locally Hermitian 1-systems of \(W_5(q)\) (Q1766241)

In this well-written and interesting note the authors prove the following two theorems: Theorem: Let \(\mathcal{M}_1\) and \(\mathcal{M}_2\) be locally Hermitian \(1\)-systems of a symplectic polar space \(W_5(q)\) with \(q > 2\) even. If \(\theta\) is an isomorphism between \(SPG(\mathcal{M}_1)\) and \(SPG(\mathcal{M}_2)\), then \(\theta\) is induced by an element \(\vartheta \in P\Gamma L(7,q)\) which maps \(cal{M}_1\) onto \(\mathcal{M}_2\). Theorem: Suppose that \(\mathcal{M}_1\) and \(\mathcal{M}_2\) are two locally Hermitian \(1\)-systems of \(W_5(q)\), \(q > 2\) even. Then the corresponding semipartial geometries \(SPG(\mathcal{M}_1)\) and \(SPG(\mathcal{M}_2)\) are isomorphic if and only if \(\mathcal{M}_1\) and \(\mathcal{M}_2\) are isomorphic for the stabilizer of \(W_5(q)\) in \(P\Gamma L(6,q)\).