Terracini matroids: algebraic matroids of secants and embedded joins

2025-11-05

AI Summary

This paper investigates the behavior of algebraic matroids under join and secant constructions of algebraic varieties, focusing on when the algebraic matroid of a join variety equals the union of the matroids of its components. Method: We introduce the novel notion of the “Terracini union matroid”, translating the classical Terracini lemma’s tangent-space analysis into a matroid-theoretic characterization of algebraic dependence. Integrating tools from algebraic geometry, matroid theory, and differential-algebraic methods, we establish a structural correspondence theorem for algebraic matroids under joins and secants. Contribution/Results: We derive necessary and sufficient conditions for the matroid union to be consistent (i.e., equal to the join’s matroid). Our framework explicitly resolves the algebraic dependence structure of toric surfaces and triple varieties, providing the first stability characterization of matroids in nonlinear gluing geometries. This yields a computationally tractable tool bridging algebraic geometry and combinatorial matroid theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Studying algebraic matroids behavior under joins and secants of varieties
• Introducing Terracini union concept for matroid structure analysis
• Applying results to understand toric surfaces and threefolds implications

Innovation

• Terracini matroids capture algebraic dependencies on varieties
• Study algebraic matroids under joins and secants operations
• Introduce Terracini union concept for matroid behavior analysis