Terracini matroids: algebraic matroids of secants and embedded joins

TL;DR

This paper explores how algebraic matroids behave under joins and secants of varieties, introducing the concept of Terracini unions to understand when matroids of joins match unions of individual matroids.

Contribution

It introduces the notion of Terracini unions of matroids, connecting algebraic dependencies with geometric joins and secants, with applications to toric varieties.

Findings

Terracini unions characterize when algebraic matroids of joins match matroid unions.

Results have implications for the study of toric surfaces and threefolds.

The work links algebraic geometry concepts with combinatorial matroid theory.

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.