Containment counterexamples for ideals of various configurations of points in PNPN

When I is the radical homogeneous ideal of a finite set of points in projective N  -space, PNPN, over a field K  , it has been conjectured that I(rN−N+1)I(rN−N+1) should be contained in IrIr for all r≥1r≥1. Recent counterexamples show that this can fail when N=r=2N=r=2. We study properties of the resulting ideals. We also show that failures occur for infinitely many r   in every characteristic p>2p>2 when N=2N=2, and we find additional positive characteristic failures when N>2N>2.