The second and the third smallest arrangements of hyperplanes in finite projective spaces

In this paper we determine for some values of d   the second and the third smallest configuration of hyperplanes in PG(N,q)PG(N,q). We present links with the unique extendability of arcs in PG(2,q)PG(2,q) and with (k,3)(k,3)-arcs having a unique trisecant.