This paper investigates the families of smooth complete intersections containing $r$-planes in projective spaces. We are going in a primitive way to shed some light on a point and an $r$-plane containing the point in a complete intersection from the viewpoint of projective geometry.
"Remarks on $r$-planes in Complete Intersections." Tokyo J. Math. 39 (2) 459 - 467, December 2016. https://doi.org/10.3836/tjm/1459367269