Terracini matroids: algebraic matroids of secants and embedded joins

Fatemeh Mohammadi; Jessica Sidman; Louis Theran

arxiv: 2511.03389 · v2 · pith:NP5ZUD7Pnew · submitted 2025-11-05 · 🧮 math.CO · cs.SC· math.AC· math.AG

Terracini matroids: algebraic matroids of secants and embedded joins

Fatemeh Mohammadi , Jessica Sidman , Louis Theran This is my paper

Pith reviewed 2026-05-18 01:20 UTC · model grok-4.3

classification 🧮 math.CO cs.SCmath.ACmath.AG

keywords algebraic matroidsTerracini unionsvariety joinssecant varietiesmatroid uniontoric varietiestangent spacesembedded joins

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The pith

A Terracini union of matroids holds exactly when the algebraic matroid of a variety join equals the matroid union of its summands.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a Terracini union of matroids as the case where algebraic dependencies among coordinates on a joined variety are exactly those obtained by taking the union of the dependencies on each summand. Algebraic matroids record polynomial relations satisfied by points on a variety, and joins and secants build new varieties from old ones. The authors tie this definition to Terracini's lemma, which describes tangent spaces of the join, and show that the matroid equality holds under the corresponding geometric condition. They then use the notion to study algebraic matroids on secants and on joins of toric surfaces and threefolds.

Core claim

We introduce Terracini unions of matroids to capture the precise situation in which the algebraic matroid of the join of two varieties equals the matroid union of the algebraic matroids of the two varieties separately. The definition is motivated by Terracini's lemma on tangent spaces, and the paper establishes that this equality holds for joins and secants when the Terracini union condition is satisfied. The results are illustrated by explicit computations for joins involving toric surfaces and threefolds.

What carries the argument

A Terracini union of matroids, the property that the algebraic matroid of a variety join coincides with the matroid union of the algebraic matroids of the summands.

If this is right

The algebraic matroid of any join or secant that satisfies the Terracini union condition can be obtained directly by unioning the matroids of the components.
Rank and independence calculations for algebraic matroids on toric threefolds reduce to the corresponding calculations on the generating surfaces.
Secant varieties inherit algebraic matroids from their base varieties precisely when the Terracini union condition holds at a general point.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the Terracini union condition can be checked on tangent spaces alone, it may give an algorithm to compute algebraic matroids of iterated joins without solving large systems of polynomials.
The same condition might extend to other matroid operations that arise when varieties are glued along lower-dimensional strata.

Load-bearing premise

The varieties under consideration must satisfy the geometric hypotheses that let Terracini's lemma relate the tangent spaces of the join to those of the individual summands.

What would settle it

Two concrete varieties whose join has an algebraic matroid that is strictly larger or smaller than the matroid union of the two summands, even though the varieties meet the tangent-space conditions of Terracini's lemma.

Figures

Figures reproduced from arXiv: 2511.03389 by Fatemeh Mohammadi, Jessica Sidman, Louis Theran.

**Figure 1.** Figure 1: The Veronese embedding of P 2 by cubics has three projections to the quadratic Veronese. △ The following theorem shows that in general the phenomenon observed in Example 3.2 exactly characterizes when the algebraic matroid of a join fails to be a Terracini union. Theorem 3.3 (Union Theorem). Let K be a field with K = K and char K = 0. If X1, . . . , Xs ⊆ A N K are irreducible affine cones with join X, then… view at source ↗

**Figure 2.** Figure 2: Monomials corresponding to the quadratic Veronese. First, we will show that U is dependent in M{2} . Define A =   zi zi+1 zi+3 zi+1 zi+2 zi+4 zi+3 zi+4 zi+5   . Substituting the corresponding monomials for the zi into A below shows that the 2 × 2 minors of A vanish on the torus embedding and hence are in I(X).   s a t b s a+1t b s a t b+1 s a+1t b s a+2t b s a+1t b+1 s a t b+1 s a+1t b+1 s a t b+2  … view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines Terracini unions of matroids to mark when the algebraic matroid of a join equals the matroid union of the pieces, using Terracini's lemma as the direct motivation, then checks the idea on toric surfaces and threefolds.

read the letter

The main takeaway is that the authors introduce the Terracini union of matroids as a way to encode the condition under which the algebraic matroid of an embedded join matches the matroid union of the summands. This is motivated straight from the tangent-space additivity in Terracini's lemma, so the central move is to turn that geometric fact into a matroid-theoretic definition. They then look at what this implies for toric surfaces and threefolds. That choice makes sense because toric varieties have explicit combinatorial descriptions that let you test whether the new notion lines up with known dependencies. The definition itself is the clearest new piece; it gives a clean language for tracking algebraic dependencies under joins and secants without having to redo tangent-space calculations each time. The toric examples serve as consistency checks rather than as the main technical engine. On the soft side, the value of the work will rest on how much the definition actually simplifies proofs or computations in those cases. If the applications mostly verify that the condition holds where Terracini's lemma already applies, the paper stays mostly definitional. That is not a flaw, but it does mean the payoff depends on whether readers find the new terminology useful for their own calculations. The citation pattern looks standard for the area, pulling in the usual references on algebraic matroids and secant varieties. This is the kind of paper that algebraic geometers and matroid theorists working on varieties with joins or secants would want to see. It is narrow enough that not everyone needs to read it, but the intersection group should find it worth their time. I would send it to referees. The idea is cleanly motivated, the authors have a clear target, and the toric checks give something concrete to evaluate even if the core remains a definition.

Referee Report

2 major / 3 minor

Summary. The paper introduces the notion of a Terracini union of matroids to characterize the precise condition under which the algebraic matroid of an embedded join (or secant) of varieties coincides with the matroid union of the algebraic matroids of the individual summands. This definition is motivated directly by the tangent-space additivity statement in Terracini's lemma. The manuscript then applies the framework to toric surfaces and threefolds, presenting these as consistency checks and illustrative examples rather than as the primary results.

Significance. If the central equivalence holds, the work supplies a clean dictionary between a classical differential-geometric condition and a purely combinatorial matroid operation. This could streamline computations of algebraic matroids for joins and secants, especially in the toric setting where explicit equations are available. The explicit link to Terracini's lemma is a strength that may encourage further cross-fertilization between algebraic geometry and matroid theory.

major comments (2)

§3 (Definition of Terracini union): The definition is stated as an if-and-only-if equivalence between the matroid-union property and the Terracini-union condition. It would strengthen the manuscript to include a short self-contained argument (or reference to a standard fact) showing necessity, i.e., that failure of tangent-space additivity forces a dependence in the join matroid that is not present in the union.
§4.2 (toric threefold examples): The claim that the Terracini-union condition holds for the listed joins of toric surfaces relies on explicit tangent-space calculations. These calculations should be recorded in an appendix or supplementary file so that the reader can verify the dimension counts without reconstructing the ambient projective space coordinates.

minor comments (3)

§1: The introduction assumes familiarity with both algebraic matroids and Terracini's lemma; a one-paragraph reminder of the latter (including the precise statement used) would improve accessibility for combinatorial readers.
Notation: The symbol for the algebraic matroid of a variety is introduced late; placing a consolidated notation table after the introduction would reduce cross-referencing.
References: Several recent papers on algebraic matroids of secants are cited only in the introduction; a dedicated subsection or paragraph comparing the present approach to those works would clarify novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report and positive assessment of the manuscript. We address each major comment below and have incorporated revisions to strengthen the exposition.

read point-by-point responses

Referee: §3 (Definition of Terracini union): The definition is stated as an if-and-only-if equivalence between the matroid-union property and the Terracini-union condition. It would strengthen the manuscript to include a short self-contained argument (or reference to a standard fact) showing necessity, i.e., that failure of tangent-space additivity forces a dependence in the join matroid that is not present in the union.

Authors: We agree that an explicit argument for the necessity direction improves clarity. In the revised manuscript we have inserted a short paragraph immediately after the definition in §3. It notes that if tangent-space additivity fails at a general point, then the differential of the join morphism has rank strictly less than the sum of the individual ranks; the resulting linear dependence among the coordinate differentials on the join is therefore not implied by the matroid union of the summands, and hence appears as an extra circuit in the algebraic matroid of the join. The argument uses only the definition of algebraic matroids via tangent spaces and the chain rule for the join map, so it is self-contained. revision: yes
Referee: §4.2 (toric threefold examples): The claim that the Terracini-union condition holds for the listed joins of toric surfaces relies on explicit tangent-space calculations. These calculations should be recorded in an appendix or supplementary file so that the reader can verify the dimension counts without reconstructing the ambient projective space coordinates.

Authors: We accept this recommendation. The revised version includes a new Appendix A that records the explicit tangent-space bases and dimension counts for each toric threefold example in §4.2. For each join we list the homogeneous coordinates of the ambient projective space, the parametrizations of the two summands, the Jacobian matrices of the individual maps, and the concatenated matrix whose rank is compared with the sum of the individual ranks. This allows direct verification of the Terracini-union condition without reconstructing the ambient spaces. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces the Terracini union of matroids as an explicit new definition motivated by the external, established Terracini's lemma in algebraic geometry. This definition is constructed to match the condition for algebraic matroids of joins equaling matroid unions of summands, but the paper presents it as a definitional tool rather than deriving a result that reduces to its own inputs by construction. Applications to toric surfaces and threefolds are described as illustrations and consistency checks, not as load-bearing steps that close a self-referential loop. No fitted parameters, self-citation chains, or ansatzes imported from prior author work are used to force the central claims. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work introduces a new definition relying on an existing geometric lemma; no free parameters or data fits are mentioned.

axioms (1)

domain assumption Terracini's lemma applies to the varieties whose algebraic matroids are under study
The Terracini union notion is explicitly motivated by the lemma, so its applicability is presupposed for the results to hold.

invented entities (1)

Terracini union of matroids no independent evidence
purpose: Captures the condition under which the algebraic matroid of a join equals the matroid union of the summands
Newly introduced concept to encode the desired coincidence property.

pith-pipeline@v0.9.0 · 5629 in / 1215 out tokens · 37551 ms · 2026-05-18T01:20:35.089490+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

[1]

Point Configurations and Cayley-Menger Varieties

[Ba˜ n+] Hector Ba˜ nos, Nathaniel Bushek, Ruth Davidson, Elizabeth Gross, Pamela Harris, Robert Krone, Colby Long, AJ Stewart, and Robert Walker.PhylogeneticTrees: invariants for group-based phylogenetic models. Version 2.0. AMacaulay2package available athttps://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/ packages. [Ba˜ n+20] Hector Ba˜ nos, Nathan...

work page internal anchor Pith review Pith/arXiv arXiv 2020
[2]

Cambridge University Press, Cambridge, 2016, pp

[EH16] David Eisenbud and Joe Harris.3264 and all that—a second course in algebraic geometry. Cambridge University Press, Cambridge, 2016, pp. xiv+616. [Ful13] William Fulton.Intersection theory. Vol

work page 2016
[3]

Characterizing Generic Global Rigidity

[GHT10] Steven J. Gortler, Alexander D. Healy, and Dylan P. Thurston. “Characterizing Generic Global Rigidity”. In:American Journal of Mathematics132.4 (2010), pp. 897–

work page 2010
[4]

Representation of matroids

[Ing71] Aubrey W. Ingleton. “Representation of matroids”. In:Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969). Academic Press, London-New York, 1971, pp. 149–167. [LMR22] Antonio Laface, Alex Massarenti, and Rick Rischter. “On secant defectiveness and identifiability of Segre–Veronese varieties”. In:Revista Matem´ atica Iberoamer...

work page arXiv 1969
[5]

The algebraic geometry of stresses in frame- works

[WW83] Neil L. White and Walter Whiteley. “The algebraic geometry of stresses in frame- works”. In:SIAM J. Algebraic Discrete Methods4.4 (1983), pp. 481–511. 15 AppendixA.Linear algebra We need the following result in combinatorial linear algebra to connect Terracini’s Lemma with the partition-based definition of matroid union. This result also captures t...

work page 1983

[1] [1]

Point Configurations and Cayley-Menger Varieties

[Ba˜ n+] Hector Ba˜ nos, Nathaniel Bushek, Ruth Davidson, Elizabeth Gross, Pamela Harris, Robert Krone, Colby Long, AJ Stewart, and Robert Walker.PhylogeneticTrees: invariants for group-based phylogenetic models. Version 2.0. AMacaulay2package available athttps://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/ packages. [Ba˜ n+20] Hector Ba˜ nos, Nathan...

work page internal anchor Pith review Pith/arXiv arXiv 2020

[2] [2]

Cambridge University Press, Cambridge, 2016, pp

[EH16] David Eisenbud and Joe Harris.3264 and all that—a second course in algebraic geometry. Cambridge University Press, Cambridge, 2016, pp. xiv+616. [Ful13] William Fulton.Intersection theory. Vol

work page 2016

[3] [3]

Characterizing Generic Global Rigidity

[GHT10] Steven J. Gortler, Alexander D. Healy, and Dylan P. Thurston. “Characterizing Generic Global Rigidity”. In:American Journal of Mathematics132.4 (2010), pp. 897–

work page 2010

[4] [4]

Representation of matroids

[Ing71] Aubrey W. Ingleton. “Representation of matroids”. In:Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969). Academic Press, London-New York, 1971, pp. 149–167. [LMR22] Antonio Laface, Alex Massarenti, and Rick Rischter. “On secant defectiveness and identifiability of Segre–Veronese varieties”. In:Revista Matem´ atica Iberoamer...

work page arXiv 1969

[5] [5]

The algebraic geometry of stresses in frame- works

[WW83] Neil L. White and Walter Whiteley. “The algebraic geometry of stresses in frame- works”. In:SIAM J. Algebraic Discrete Methods4.4 (1983), pp. 481–511. 15 AppendixA.Linear algebra We need the following result in combinatorial linear algebra to connect Terracini’s Lemma with the partition-based definition of matroid union. This result also captures t...

work page 1983