Matroid Varieties in Algebraic Geometry

• Matroid varieties are algebraic varieties that encode the combinatorial dependencies of matroids through their realization spaces and Zariski closures.
• Their defining ideals are built from circuit, Grassmann–Cayley, and liftability polynomials, which translate point-line incidences and collinearity into algebraic relations.
• The study of these varieties bridges classical geometric theorems and modern computational challenges, offering tools for moduli analysis and stratification in combinatorial algebraic geometry.

Matroid varieties are algebraic geometric models that encapsulate the combinatorial structure of matroids via realization spaces and their Zariski closures. These varieties serve as fundamental objects of paper in combinatorial algebraic geometry, bridging the gap between abstract matroid theory and classical projective geometry. Recent developments have focused on defining equations, irreducibility, and combinatorial decompositions for key families, notably point-line configurations associated to classical combinatorial theorems (such as theorems of Pascal and Pappus) and broader classes such as cactus matroids.

1. Definition and Realization of Matroid Varieties

For a simple rank-three matroid $M$ on a ground set $[d]$, a realization is a tuple of vectors $$\gamma = (\gamma_1, \ldots, \gamma_d)\subset\mathbb{C}^3$$ such that a subset $\{i_1, \ldots, i_p\}$ is dependent in $M$ if and only if $\gamma_{i_1},\ldots,\gamma_{i_p}$ are linearly dependent in $\mathbb{C}^3$. The realization space $\Gamma_M$ is the subset of $\mathbb{C}^{3d}$ parameterizing such configurations. The Zariski closure $V_M := \overline{\Gamma_M}$ is called the matroid variety of $M$.

The matroid variety $V_M$ structurally encodes the dependencies of $M$ in a geometric fashion. In the context of point-line configurations (rank-3 matroids), $V_M$ is governed by the arrangement of points and their specified collinearities.

2. Generating Sets and Defining Ideals

The defining ideal $I_M$ of a matroid variety is constructed from three main types of generators:

• Circuit polynomials (circuit ideal $I_{\mathcal{C}(M)}$): For each circuit (minimal dependent subset) $\{i,j,k\}$, the bracket polynomial $[ijk]$ (the determinant/minor for columns $i,j,k$) vanishes on all realizations. The set $\{[ijk]:\{i,j,k\}\in\mathcal{C}(M)\}$ spans the circuit ideal.
• Grassmann–Cayley polynomials ($G_M$): These encode concurrency conditions (when three lines are concurrent in $\mathbb{P}^2$) and arise from bracket combinations derived via Grassmann–Cayley algebra. For instance, concurrent lines may yield relations like $[123][456]-[124][356]$.
• Lifting polynomials ($I_M^{(\text{lift})}$): Based on “liftability,” one forms a liftability matrix $\mathcal{M}_q(M)$ (rows index circuits of $M$; columns are points), where the $(i,j)$-th entry is $(-1)^{i-1} [c_1,\dots,\hat{c}_i,\dots,c_n,q]$ for a circuit $c$ and a fixed vector $q$ off a hyperplane. Suitable minors of this matrix yield polynomials vanishing on $V_M$.

The main result is that $I_M$ can be generated—up to radical—by these three classes:

$$I_M = \sqrt{I_{\mathcal{C}(M)} + G_M + I_M^{(\text{lift})}}.$$

For small or particularly nice families (such as cactus, Pascal, or Pappus matroids), finite generating sets have been computed explicitly, demonstrating feasibility for configurations of moderate size (Liwski et al., 9 Jun 2025).

3. Irreducibility and Structural Properties

Irreducibility of $V_M$ is a central question. For cactus matroids—matroids whose underlying dependency graph is a cactus graph (every two cycles share at most one vertex)—realizability and irreducibility have been proved. The concept of nilpotence is used: a matroid is nilpotent if, by iteratively restricting to points of degree at least two, one eventually reduces to the empty matroid. Cactus configurations are shown to be nilpotent, and thus every cactus matroid has an irreducible matroid variety (Liwski et al., 9 Jun 2025).

More generally, for other point-line configurations:

• Liftable configurations (where degenerate collinear configurations can be non-degenerately “lifted” to the matroid) yield irreducible matroid varieties.
• Quasi-liftable configurations (not fully liftable, but every proper subconfiguration obtained by line removal is liftable) admit a decomposition of their circuit variety into a “degenerate” (all-point-on-a-line) component plus the non-degenerate component.

For classical configurations such as Pascal and Pappus, explicit finiteness and irreducibility claims for $V_M$ are established, though the saturation step necessary to incorporate all independence relations is computationally intensive (Liwski et al., 9 Jun 2025).

4. Circuit Varieties and Irreducible Decompositions

Circuit varieties $V_{\mathcal{C}(M)}$, defined as the common zero locus of the circuit ideal, stratify the solution space by combinatorial degenerations. For cactus configurations, the irreducible components correspond to switching certain points (from the set $Q_M$ of points contained in at least three lines) to loops:

$$V_{\mathcal{C}(M)} = \bigcup_{J\subseteq Q_M} V_{M(J)}$$

where $M(J)$ is obtained from $M$ by making points in $J$ into loops (degenerate points). Hence, there are at most $2^{|Q_M|}$ components (Liwski et al., 9 Jun 2025). This decomposition is tightly controlled by the combinatorics of $M$ and reflects the possible degeneracies in realizations.

A table illustrates this for some classical matroids:

Matroid	# Circuits	$|Q_M|$	Max # Components
Pascal	7	1	2
Pappus	9	3	8
Cactus (general)	$k$	$m$	$2^m$

5. Grassmann–Cayley Algebra and Geometric Methods

Geometric liftability and Grassmann–Cayley algebra underpin the construction of polynomials in $G_M$. The Grassmann–Cayley algebra encodes geometric theorems (concurrency, collinearity) as bracket identities (e.g., $[123][456] - [124][356]$ corresponds to the concurrency of three lines defined by points $1,2,3$, $4,5,6$, etc.). For configurations such as Pascal or Pappus, explicit bracket additions derived from geometric theorems (such as Pascal’s collinearity of points of intersection on a conic, or Pappus’s configuration) are included in $G_M$.

Liftability polynomials are constructed by studying when tuples of collinear points can be lifted off a degenerate line while preserving the matroid’s incidence relations. The vanishing of certain minors in the liftability matrix is both a certificate of non-liftability and a generator for $I_M^{(\text{lift})}$.

6. Computability and Computational Challenges

While the general process for generating $I_M$ is conceptually clear, in practice the computation is limited by the combinatorial growth induced by saturation over all independence relations, particularly in matroids of larger size. The Pascal and Pappus matroids serve as benchmarks: for Pascal, the generating set includes 7 circuits, 7 Grassmann–Cayley polynomials, and over 700,000 lifting polynomials; for Pappus, analogous numbers are even larger. Full saturation is currently feasible only for very small matroids (Liwski et al., 9 Jun 2025).

7. Connections and Implications

The identification and paper of matroid varieties for cactus, Pascal, and Pappus matroids reinforce the deep interplay between combinatorial geometry and algebraic geometry. The explicit bridge between the combinatorics of matroids (circuits, dependencies, graph-theoretic properties) and algebraic invariants of their realization spaces enables rigorous investigation of questions in rigidity theory, moduli problems, and determinantal geometry.

Moreover, the explicit combinatorial control over irreducible components in the circuit variety provides practical tools for stratification analysis in more general classes of matroid-related moduli spaces. These methods generalize classical geometric incidence theorems, support practical computations in small cases, and highlight the intersection of algebraic and combinatorial methods in matroid theory.

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References: The above draws on (Liwski et al., 9 Jun 2025), which provides explicit generating sets and decompositions for Pascal, Pappus, and cactus matroid varieties; the irreducibility criterion and realization methods are detailed using nilpotence, Grassmann–Cayley algebra, and liftability techniques as established in that work.