Algebraic polygons
Let \({\mathfrak P}= ({\mathcal P}, {\mathcal L}, {\mathcal F})\) be a generalized \(n\)-gon \(n\geq 3\), and let \(d:{\mathcal P} \cup{\mathcal L} \to\{0, \dots,n\}\) be the distance function of \({\mathfrak P}\). Assume that \({\mathcal P}\) and \({\mathcal L}\) are \(K\)-varieties for some algebraically closed field \(K\). Then \({\mathfrak P}\) is called an algebraic polygon if the set \(D=\{(x,y)\in ({\mathcal P} \cup{\mathcal L}) \times ({\mathcal P} \cup{\mathcal L}) |d(x,y)= n-1\}\) is locally closed and \(f_{n-1}: D\to{\mathcal P} \cup {\mathcal L}\) is continuous, where \(f_{n-1} (x,y)=z\) if \(d(x,z)=n-2\) and \(d(y,z)=1\). The authors show that if \(K\) is of characteristic 0 then there are precisely 3 examples up to duality, namely the projective plane, the symplectic quadrangle, and the split Cayley hexagon over \(K\). The proof is achieved by using model theoretic concepts which allow the transfer of the problem from \(K\) to the field \(\mathbb{C}\) of complex numbers; in this particular case the theorem has already been proved by the first author in Math. Z. 223, No. 2, 333-341 (1996). As a corollary to this remarkable result one gets that the only irreducible spherical Tits systems with closed Borel subgroups in connected algebraic groups over algebraically closed fields of characteristic 0 are the classical ones.
scientific article; zbMATH DE number 922313