  We compare and contrast results of E. Davis, of A. Bigatti, A.V. Geramita and
the author, and of J. Ahn and the author. The underlying idea is that certain
numerical conditions on the Hilbert function of a finite set of points in
projective space force geometric consequences on the base loci of the linear
systems of hypersurfaces containing the points. When the points have uniform
position, this tends to force better behavior for these base loci. However,
unexpected behavior is still possible, and we give examples.
