Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular, many of the exceptional Lie incidence geometries occur. In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected case of some other well-known characterizations.

We study to which extent the family of pairs of subspaces of a vector space related to each other via intersection properties determines the vector space. In another language, we study to which extent the family of vertices of the building of a projective space related to each other via several natural respective conditions involving the Weyl distance and incidence determines the building. These results can be seen as generalizations of and variations on the Fundamental Theorem of Projective Geometry.

Hjelmslev–Moufang (HM) planes are point-line geometries related to the exceptional algebraic groups of type $\mathsf{E}_6$. More generally, point-line geometries related to spherical Tits buildings—Lie incidence geometries—are the prominent examples of parapolar spaces: axiomatically defined geometries consisting of points, lines and symplecta (structures isomorphic to polar spaces). In this paper we classify the parapolar spaces with a similar behaviour as the HM planes, in the sense that their symplecta never have a non-empty intersection. Under standard assumptions, we obtain that the only such parapolar spaces are exactly given by the HM planes and their close relatives (arising from taking certain restrictions). On the one hand, this work complements the algebraic approach to HM planes using Jordan algebras and due to Faulkner in his book ‘The Role of Nonassociative Algebra in Projective Geometry’, published by the American Mathematical Society in 2014; on the other hand, it provides a new tool for classification and characterization problems in the general theory of parapolar spaces.

We provide a common description and geometric characterization of the Veronesean representations of all projective spaces defined over finite-dimensional quadratic alternative division algebras. We also study the homogeneity of these representations, and the irreducibility and indecomposability of the induced projective representations of the corresponding simple (automorphism) groups.

Let Ω i and Ω j be the sets of elements of respective types i and j of a polar space ∆ of rank at least 3, viewed as a Tits-building. For any Weyl distance δ between Ω i and Ω j , we show that δ is characterised by i and j and two additional numerical parameters k and . We consider permutations ρ of Ω i ∪ Ω j that preserve a single Weyl distance δ. Up to a minor technical condition on , we prove that, up to trivial cases and two classes of true exceptions, ρ is induced by an automorphism of the Tits-building associated to ∆, which is always a type-preserving automorphism of ∆ (and hence preserving all Weyl-distances), unless ∆ is hyperbolic, in which case there are outer automorphisms. For each class of exceptions, we determine a Tits-building ∆ in which ∆ naturally embeds and is such that ρ is induced by an automorphism of ∆ . At the same time, we prove similar results for permutations preserving a natural incidence condition. These yield combinatorial characterisations of all groups of algebraic origin which are the full automorphism group of some polar space as the automorphism group of many bipartite graphs.(PS3) For U ∈ Ω with dim(U) = n − 1 and p ∈ X \ U, the union of all elements of Ω of dimension 1 containing p and intersecting U nontrivially is an element of Ω of dimension n − 1 which intersects U in a hyperplane.(PS4) There are two disjoint elements of Ω of dimension n − 1.A set X of cardinality at least two, together with Ω = X ∪ { } is considered to be a polar space of rank 1. Henceforth, ∆ denotes a polar space of rank n with n ≥ 2.Collinearity and opposition − An element of Ω of dimension n − 1 is called a maximal singular subspace (MSS for short) and an element of Ω of dimension 1 is called a line. Let x and y be two distinct points. If they are on a common line, they are called collinear and we write x ⊥ y, if not, they are called opposite. The set of points equal or collinear with x is denoted by x ⊥ . A subspace S of ∆ is a subset of X such that the lines joining any two collinear points of S are contained in S. Moreover, if S contains no pair of opposite points, the subspace is called singular. The elements of Ω are precisely the singular subspaces of ∆. If U and V are singular subspaces with U ⊆ V , then the codimensionFor a singular subspace U, we define U ⊥ as x∈U x ⊥ . For any singular subspace V , we say that U and V are collinear if V ⊆ U ⊥ . If they are collinear but disjoint, we write U ⊥ V . Let T be a set of pairwise collinear singular subspaces. We denote by 〈T 〉 the smallest singular subspace containing all members of T , and we also say that the members of T generate 〈T 〉 or that 〈T 〉 is spanned by the members of T . If T consists of two distinct collinear points x, y, we denote the unique line joining these points by xy. The projection proj V (U) of a singular subspace U on a singular subspace V is V ∩ U ⊥ and the subspace spanned by U and proj V (U) is denoted byis empty, we say that U and V are semi-opposite. Now let U and V be semi-opposite singular subspaces. If dim(U) = dim(V...