×

Quadric Veronesean caps. (English) Zbl 1271.51001

Summary: In 2008, E. Ferrara Dentice and G. Marino [Discrete Math. 308, No. 2-3, 299–302 (2008; Zbl 1151.51011)] provided a characterization theorem for Veronesean caps in \(\mathrm{PG}(N,\mathbb{K})\), with \(\mathbb{K}\) a skewfield. This result extends the theorem for the finite case proved by J.A. Thas and the second author in [Q. J. Math. 55, No. 1, 99–113 (2004; Zbl 1079.51004)]. However, although the statement of this theorem is correct, the proof given by Ferrara Dentice and Marino [loc. cit.] is incomplete, as they borrow some lemmas from the paper by J. A. Thas and the second author [loc. cit.], which are proved using counting arguments and hence require a different approach in the infinite case. In this paper, we use the Veblen-Young theorem to fill these gaps. Moreover, we then use this classification of Veronesean caps to provide a further general geometric characterization.

MSC:

51A45 Incidence structures embeddable into projective geometries
51M35 Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations)