Combinatorial Veronese structures, their geometry, and problems of embeddability (Q927815)

Let \(X\) be an arbitrary set. The point set of a Veronese structure \(\mathbf{V}_k(X)\) over \(X\) consists of all \(k\)-multisets with elements from \(X\). The authors present a detailed description of the geometry of the smallest reasonable Veronese structure \(\mathbf{V}_3(3)\) (various interpretations and representations, configurational axioms, realizability in a projective space, self-duality). It is shown that none of the configurations \(\mathbf{V}_k(n)\) and their duals, except \(\mathbf{V}_3(3)\), is embeddable into a Desarguesian projective space. This yields a large class of Desarguesian partial linear spaces which cannot be embedded into a Desarguesian projective space. Quite surprisingly, it turns out that some of these configurations can be completed to a linear space.