Veronesean representations of Moufang planes

Abstract

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).

Veronese Representation of Projective Hjelmslev Planes Over Some Quadratic Alternative Algebras

“…Since V * is not collinear to x, V * and Π Y x are complementary subspaces of Y by Lemma 8.3(ii) and (iv). In the Veronese variety (ρ(X), ρ(Ξ)), the point ρ(x) is not contained in ρ(C * ) (since x is not collinear to V * ), so ρ(T x ) = T ρ(x) and ρ(C * ) are also complementary subspaces by the properties of Veronese varieties (this can be verified algebraically but it has also been proven in Proposition 4.5 of [6]). Since T x ∩ Y and C * ∩ Y are complementary subspaces of Y and since the projections ρ(T x ) and ρ( C * ) from Y onto F are complementary in F , we obtain that T x and C * are complementary in Y, F = PG(N, K).…”

“…Since 6, 8 do not occur in any such triple, the remaining blocks are [1,6] and [1,8], and they need to be distinct (there is no * -triple neither containing 1 nor 6 nor 8). Hence the block [1,6] has to contain x-triples not containing 1 and 6, but there are exactly three such. Consequently, there is only one possibility: [1,6] A straightforward verification shows that each such block sums up to zero.…”

“…Hence the block [1,6] has to contain x-triples not containing 1 and 6, but there are exactly three such. Consequently, there is only one possibility: [1,6] A straightforward verification shows that each such block sums up to zero.…”

“…The quadrics are hyperboloids, ellipsoids and quadratic cones, respectively. In [6] and [12], the former two have been extended to planes over higher-dimensional non-degenerate quadratic alternative algebras over K, which geometrically corresponds to using higher-dimensional quadrics, namely hyperbolic quadrics (nondegenerate quadrics of maximal Witt index) and elliptic quadrics (nondegenerate quadrics of Witt index 1).…”

Veronesean representations of projective spaces over quadratic associative division algebras

“…It is well known that, to every quadratic alternative division algebra A over the field K, corresponds a Veronesean representation of the projective plane associated with A in a projective space over K of dimension 3d + 2, where d = dim K A (see for instance [9]). Such Veronesean representations realise in a geometric way the Tits indices 2…”