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In 1901 Severi [18] proved that the complex quadric Veronese variety is determined by three algebraic/differential geometric properties. In 1984 Mazzocca and Melone [10] obtained a combinatorial analogue of this result for finite quadric Veronese varieties. We make further abstraction of these properties to characterize Veronesean representations of all the Moufang projective planes defined over a quadratic alternative division algebra over an arbitrary field. In the process, new Veroneseans over a nonperfect field of characteristic 2 (related to purely inseparable field extensions) are found, and their corresponding projective representations of the associated groups studied. We show that these representations are indecomposable, but reducible, and determine their (irreducible) quotient and kernel. P 2 (A) with A a quadratic alternative division ring are all Moufang planes. Conversely, every Moufang plane is isomorphic to such a P 2 (A) or to a projective plane coordinatized by a skew field.Concerning projective spaces P(V ), with V a right vector space over some skew field K, we will denote the subspace spanned by a (point) set S with S . We usually view subspaces as sets of points. Also, we will need to consider projections. These are maps with a projection center U and an image W , where U and W are complementary subspaces, that is, U ∩ W = ∅ and U, W is the whole projective space. The projection from U to W maps a point p / ∈ U to the point U, p ∩ W . The projection from U onto W is not defined on the points of U , and if we consider a subspace S intersecting U , then by the projection of S we mean the subspace U, S ∩ W (so for subspaces, we do not project pointwise since otherwise we always have to avoid the points of U ∩ S, which only makes the notation cumbersome).We end this introduction by mentioning that large parts of Section 6 are taken from the PhD thesis of the first author, who obtained a partial characterization of V 2 (K, A) (mainly for the case of char K = 2).

Let $\Pi = (P,L,I)$ denote a rank two geometry. In this paper, we are interested in the largest value of $|X||Y|$ where $X \subset P$ and $Y \subset L$ are sets such that $(X \times Y) \cap I = \emptyset$. Let $\alpha(\Pi)$ denote this value. We concentrate on the case where $P$ is the point set of $\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\mathsf{PG}(n,q)$. In the case that $\Pi$ is the projective plane $\mathsf{PG}(2,q)$, where $P$ is the set of points and $L$ is the set of lines of the projective plane, Haemers proved that maximal arcs in projective planes together with the set of lines not intersecting the maximal arc determine $\alpha(\mathsf{PG}(2,q))$ when $q$ is an even power of $2$. Therefore, in those cases,\[ \alpha(\Pi) = q(q - \sqrt{q} + 1)^2.\] We give both a short combinatorial proof and a linear algebraic proof of this result, and consider the analogous problem in generalized polygons. More generally, if $P$ is the point set of $\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\mathsf{PG}(n,q)$, where $1 \leq k \leq n - 1$, and $\Pi_q = (P,L,I)$, then we show as $q \rightarrow \infty$ that \[ \frac{1}{4}q^{(k + 2)(n - k)} \lesssim \alpha(\Pi) \lesssim q^{(k + 2)(n - k)}.\] The upper bounds are proved by combinatorial and spectral techniques. This leaves the open question as to the smallest possible value of $\alpha(\Pi)$ for each value of $k$. We prove that if for each $N \in \mathbb N$, $\Pi_N$ is a partial linear space with $N$ points and $N$ lines, then $\alpha(\Pi_N) \gtrsim \frac{1}{e}N^{3/2}$ as $N \rightarrow \infty$.

We provide a uniform framework to study the exceptional homogeneous compact geometries of type C3. This framework is then used to show that these are simply connected, answering a question by Kramer and Lytchak, and to calculate the full automorphism groups.MSC 2010: 51E24, 57S15