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Terracini matroids: algebraic matroids of secants and embedded joins

Fatemeh Mohammadi, Jessica Sidman, Louis Theran

TL;DR

The paper develops a matroidal framework for understanding how algebraic dependencies among coordinates behave under joins and secants of irreducible varieties. It introduces the Terracini union concept and proves two key theorems: the Sub-union Theorem, which gives a natural upper bound $M(X_1+\cdots+X_s) \preceq M(X_1)\vee\cdots\vee M(X_s)$, and the Union Theorem, which characterizes when equality holds via projections and defectivity of projected joins. The results are applied to curves and toric varieties, revealing that equality is subtle and can fail due to combinatorial obstructions not detected by tangent-space methods. The work connects algebraic matroids with geometric notions of secant varieties and rigidity, and poses open questions about deeper structural aspects and potential higher-fold Terracini decompositions.

Abstract

Applications of algebraic geometry have sparked much recent work on algebraic matroids. An algebraic matroid encodes algebraic dependencies among coordinate functions on a variety. We study the behavior of algebraic matroids under joins and secants of varieties. Motivated by Terracini's lemma, we introduce the notion of a Terracini union of matroids, which captures when the algebraic matroid of a join coincides with the matroid union of the algebraic matroids of its summands. We illustrate applications of our results with a discussion of the implications for toric surfaces and threefolds.

Terracini matroids: algebraic matroids of secants and embedded joins

TL;DR

The paper develops a matroidal framework for understanding how algebraic dependencies among coordinates behave under joins and secants of irreducible varieties. It introduces the Terracini union concept and proves two key theorems: the Sub-union Theorem, which gives a natural upper bound , and the Union Theorem, which characterizes when equality holds via projections and defectivity of projected joins. The results are applied to curves and toric varieties, revealing that equality is subtle and can fail due to combinatorial obstructions not detected by tangent-space methods. The work connects algebraic matroids with geometric notions of secant varieties and rigidity, and poses open questions about deeper structural aspects and potential higher-fold Terracini decompositions.

Abstract

Applications of algebraic geometry have sparked much recent work on algebraic matroids. An algebraic matroid encodes algebraic dependencies among coordinate functions on a variety. We study the behavior of algebraic matroids under joins and secants of varieties. Motivated by Terracini's lemma, we introduce the notion of a Terracini union of matroids, which captures when the algebraic matroid of a join coincides with the matroid union of the algebraic matroids of its summands. We illustrate applications of our results with a discussion of the implications for toric surfaces and threefolds.

Paper Structure

This paper contains 13 sections, 14 theorems, 42 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Secants and joins
  4. Matroids
  5. Algebraic matroids
  6. Algebraic matroids and secant varieties
  7. Applications
  8. Curves
  9. Toric varieties
  10. Comparison with tangent-space methods
  11. Open questions
  12. Linear algebra
  13. Loops and coloops

Key Result

Theorem A

Let $K$ be an algebraically closed field of characteristic zero. If $X_1, \ldots, X_s\subseteq \mathbb{A}_K^N$ are irreducible affine cones with join $X$, then

Figures (2)

  • Figure 1: The Veronese embedding of $\mathbb{P}^2$ by cubics has three projections to the quadratic Veronese.
  • Figure 2: Monomials corresponding to the quadratic Veronese.

Theorems & Definitions (39)

  • Example 1.2: Mixture model
  • Example 1.3
  • Theorem A: Sub-union Theorem
  • Theorem B: Union Theorem
  • Example 1.4: Generic rigidity
  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • proof
  • Example 2.4
  • ...and 29 more