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Apolarity for border cactus decompositions

Weronika Buczyńska, Jarosław Buczyński

TL;DR

This work extends border apolarity from secant varieties to border cactus decompositions on smooth toric varieties over arbitrary algebraically closed fields by employing the Cox ring and multigraded Hilbert schemes. It defines border cactus witnesses as homogeneous ideals in the Cox ring that sit inside the apolar annihilator of a given form, and links these witnesses to Hilbert-function data through a precise comparison between the usual and multigraded Hilbert schemes. The main contributions are a general weak border apolarity theorem and an if-and-only-if ABCD framework: (i) a witness-ideals existence with degreewise codimension constraints for $F\in \sigma_r(X)$ and (ii) a classification of border cactus decompositions via a finite collection of Hilbert functions and birational Hilbert-scheme correspondences. The paper also develops the Grassmann-relative linear-span machinery to realize and study linear spans of cactus configurations in a toric context, and situates these constructions within the broader literature on cactus varieties and apolarity, providing tools that apply uniformly across fields and incorporate linear subspace cases. These results deepen the algebraic-geometric understanding of cactus decompositions, offering a robust framework for witness computation and structural analysis via the interplay of Hilbert schemes, Cox rings, and Grassmannians.

Abstract

The border apolarity technique was introduced in our earlier work for secant varieties over complex numbers. We extend the theory to cactus varieties of toric varieties over any algebraically closed field. A border cactus decomposition is a mulithomogeneous ideal in the Cox ring (also called the total coordinate ring) of the toric variety that witnesses that a given point is in a specific cactus variety. The definition of such witness uses apolarity and we describe the set of ideals that are credible witnesses for this purpose in terms of a correspondence between the usual Hilbert scheme (parametrising all closed subschemes of the toric variety) and the multigraded Hilbert scheme (parametrising all multihomogeneous ideals in the Cox ring). We also take this opportunity to extend the border apolarity to linear subspaces (in non-border setting, this is equivalent to simultaneous decompositions).

Apolarity for border cactus decompositions

TL;DR

This work extends border apolarity from secant varieties to border cactus decompositions on smooth toric varieties over arbitrary algebraically closed fields by employing the Cox ring and multigraded Hilbert schemes. It defines border cactus witnesses as homogeneous ideals in the Cox ring that sit inside the apolar annihilator of a given form, and links these witnesses to Hilbert-function data through a precise comparison between the usual and multigraded Hilbert schemes. The main contributions are a general weak border apolarity theorem and an if-and-only-if ABCD framework: (i) a witness-ideals existence with degreewise codimension constraints for and (ii) a classification of border cactus decompositions via a finite collection of Hilbert functions and birational Hilbert-scheme correspondences. The paper also develops the Grassmann-relative linear-span machinery to realize and study linear spans of cactus configurations in a toric context, and situates these constructions within the broader literature on cactus varieties and apolarity, providing tools that apply uniformly across fields and incorporate linear subspace cases. These results deepen the algebraic-geometric understanding of cactus decompositions, offering a robust framework for witness computation and structural analysis via the interplay of Hilbert schemes, Cox rings, and Grassmannians.

Abstract

The border apolarity technique was introduced in our earlier work for secant varieties over complex numbers. We extend the theory to cactus varieties of toric varieties over any algebraically closed field. A border cactus decomposition is a mulithomogeneous ideal in the Cox ring (also called the total coordinate ring) of the toric variety that witnesses that a given point is in a specific cactus variety. The definition of such witness uses apolarity and we describe the set of ideals that are credible witnesses for this purpose in terms of a correspondence between the usual Hilbert scheme (parametrising all closed subschemes of the toric variety) and the multigraded Hilbert scheme (parametrising all multihomogeneous ideals in the Cox ring). We also take this opportunity to extend the border apolarity to linear subspaces (in non-border setting, this is equivalent to simultaneous decompositions).
Paper Structure (23 sections, 31 theorems, 36 equations)

This paper contains 23 sections, 31 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Motivation and historical remarks about secants
  3. Rank, decompositions, and cactus varieties
  4. Apolarity: towards border decompositions
  5. Main results
  6. Multihomogeneous ideals, saturation, and toric varieties
  7. Openness of saturation and generic fibre
  8. Reduction to affine $U$.
  9. Reduction to affine ${\mathcal{Z}}$.
  10. Translating to algebraic condition.
  11. Proving the algebraic conditon.
  12. Smooth projective toric varieties
  13. A family has finitely many Hilbert functions
  14. Hilbert schemes, Grassmann relative linear span and cactus varieties
  15. Grothendieck and multigraded Hilbert schemes
  16. ...and 8 more sections

Key Result

Theorem 1.1

Suppose $F\in \sigma_r(X)$. Then there exists a homogeneous ideal $I \subset S$ such that: Moreover, if $B$ is a connected solvable group acting on $X$ and preserving $F$, then there exists an $I$ as above which in addition is invariant under $B$.

Theorems & Definitions (36)

  • Theorem 1.1: Weak border apolarity
  • Theorem 1.2: Weak Apolarity for Border Cactus Decompositions, or weak ABCD
  • Proposition 1.3
  • Theorem 1.4: Apolarity for Border Cactus Decompositions, or ABCD
  • Theorem 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.6
  • Remark 2.7
  • ...and 26 more