A note on embedding the 4-arcs of Fano plane with quadric and cubic Veroneseans to projective spaces

The 3−spaces generate intersect pairwise in at least a line (each pair of 4−sets share two points). But since the union of two such 4-sets is a 6-set, this union generates the whole space, so they pairwise intersect in a line. Dually, the lines that we consider have empty intersection. Other examples of such sets are spreads of lines, or spreads of any kind. In this work, we present SCID properties spaces generated by 4-arcs of Fano plane.