Generalized Veronesean embeddings of projective spaces

Abstract

We classify all embeddings theta: PG(n, q) -> PG(d, q), with d >= n(n+3)/2 such that theta maps the set of points of each line to a set of coplanar points and such that the image of theta generates PG(d, q). It turns out that d = 1/2n(n+3) and all examples are related to the quadric Veronesean of PG(n, q) in PG(d, q) and its projections from subspaces of PG(d, q) generated by sub-Veroneseans (the point sets corresponding to subspaces of PG(n, q)). With an additional condition we generalize this result to the infinite case as well

Characterizations of Veronese and Segre varieties

“…These objects are natural generalizations of normal rational cubic scrolls. In [12] even more complex objects are characterized, namely, unions of such projections. The main idea of this characterization is to replace the axiom that the lines of PG(n, q) form ovals in planes of PG(m, q) by the assumption that the points of these lines are just planar sets.…”

Projective planes over 2-dimensional quadratic algebras

“…It is easy to see, since |K| > 2, see also Lemma 3.1 of [19], that all these affine lines are contained in a unique affine plane α 1 . Also, if two lines of α intersect x, z 1 in the same point, then it is obvious that the corresponding lines in α 1 are parallel, and so the line x, z 1 of α corresponds to the line at infinity of α 1 .…”

“…This yields a generalized Veronesean embedding of the projective completion of G y in α, β , provided we let y play the role of the unique point at infinity. The Main Result-General Version of Section 5 of [19] implies that the projection of X y from y onto A is a normal rational cubic scroll. Hence, since X y consists of the union of lines through y, the intersection of X y with A is itself a normal rational cubic scroll, finalizing the proof of the proposition.…”

Some results on caps and codes related to orthogonal Grassmannians — a preview

“…Following [20], (see also [5]), when condition (2) is replaced by (2') e maps any line of Γ onto a non-singular conic of PG(V ), we say that e is a Veronese embedding of Γ.…”