On unions of scrolls along linear spaces

According to the classification resulting from the successive contributions by Bertini, Del Pezzo and Xambó, the equidimensional varieties of minimal degree which are connected in codimension one are of three types: quadric hypersurfaces, cones over the Veronese surface in P5 and unions of scrolls embedded in linear subspaces. In this paper we give a complete constructive characterization of the ideals defining varieties of the latter type, which were considered by Xambó. We also show that for these varieties, equidimensionality and minimal degree imply connectivity in codimension one, which provides a better understanding of the results proven by Xambó. Finally we give a complete description of all rulings of a scroll. Throughout the paper we deal with projective varieties not contained in any hyperplane.