Identifying limits of ideals of points in the case of projective space

TL;DR

This paper investigates the boundary of radical ideals in multigraded Hilbert schemes for points in projective space, providing conditions for inclusion in the closure and linking to polynomial rank theory.

Contribution

It offers new criteria for when ideals lie in the closure of radical ideals in multigraded Hilbert schemes, extending understanding in higher-dimensional projective spaces.

Findings

Sufficient condition for ideals in the closure in projective plane

Necessary condition for ideals in higher-dimensional projective space

Connection established between Hilbert schemes and polynomial rank theory

Abstract

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.