Spaces of Pencils, Grassmann Spaces, and Generalized Veronese Spaces

In this paper we construct several examples of partial linear spaces. First, we define two algebraic structures, namely the spaces of k-pencils and Grassmann spaces for vector spaces over an arbitrary field. Then we introduce the notion of generalized Veronese spaces following the definition presented in the paper [8] by Naumowicz and Prażmowski. For all spaces defined, we state the conditions under which they are not degenerated to a single line.