Quadric Veronesean Caps

Abstract

In 2008, Ferrara Dentice and Marino provided a characterization theorem for Veronesean caps in $\mathsf{PG}(N,\mathbb{K})$, with $\mathbb{K}$ a skewfield. This result extends the theorem for the finite case proved by J.A. Thas and Van Maldeghem in 2004. However, although the statement of this theorem is correct, the proof given by Ferrara Dentice and Marino is incomplete, as they borrow some lemmas from the paper of J.A. Thas and Van Maldeghem, 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.