  We study the closure of the locus of radical ideals in the multigraded
Hilbert scheme associated with a standard graded polynomial ring and the
Hilbert function of a homogeneous coordinate ring of points in general position
in projective space. In the case of projective plane, we give a sufficient
condition for an ideal to be in the closure of the locus of radical ideals. For
projective space of arbitrary dimension we present a necessary condition. The
paper is motivated by the border apolarity lemma which connects such
multigraded Hilbert schemes with the theory of ranks of polynomials.
