Limits of saturated ideals

TL;DR

This work develops a deformation-theoretic framework to decide when a homogeneous ideal is a limit of saturated ideals, introducing the fiber obstruction group $\mathrm{Ob}_{\mathrm{fiber}}(I,J)$ and stickiness concepts. It establishes openness of saturation in multigraded Hilbert schemes, provides explicit criteria for non-saturability and saturation along families, and specializes to toric and projective-space settings to obtain concrete smoothness results for the saturated locus. The methodology yields practical, computable tests for saturability and delivers applications to border apolarity and VSP, notably giving a bound that rules out certain wild polynomials in three variables and advancing lower bounds on border ranks. Collectively, the paper connects Hilbert-scheme geometry, Macaulay inverse systems, and border-rank problems to provide new tools for analyzing limits of saturated ideals and their role in tensor decompositions.

Abstract

We investigate the question whether a given homogeneous ideal is a limit of saturated ones. We provide cohomological necessary criteria for this to hold and apply them to a range of examples. Our motivation comes from the theory of border apolarity and varieties of sums of powers, where the question above is tightly connected to proving new lower bounds for border ranks of tensors.

Figures (1)

• Figure 1: Deformations of $K$ and $J$, but not yet a deformation of a pair $K\subseteq J$.

Theorems & Definitions (97)

• Theorem 1.2: Theorem \ref{['ref:deformsWhenObFibVanishes:thm']}
• Theorem 1.3: Theorem \ref{['ref:deformWhenTangentSurjective:thm']}
• Theorem 1.4: openness of saturation, Theorem \ref{['ref:opennessOfSaturation:thm']}
• Theorem 1.5: Theorem \ref{['ref:partialToGrothendieck:prop']}
• Theorem 1.6: Theorems \ref{['ref:smoothPoints:thm']}-\ref{['ref:usualClassesSmooth:cor']}
• Proposition 1.7: Proposition \ref{['prop:no_wild_polynomials']}
• Lemma 2.1
• proof
• Proposition 2.2
• proof