Generalized Uniform Matroidal Configurations

• Generalized uniform matroidal configurations are algebraic-geometric constructs that extend uniform matroids and star configurations via specialized combinatorial ideals.
• They provide explicit containment criteria for symbolic powers, using formulas for resurgence numbers and Waldschmidt constants to quantify ideal behavior.
• Their study yields practical insights into linear free resolutions, Cohen–Macaulayness, and connections to Stanley–Reisner ideals in projective settings.

A generalized uniform matroidal configuration is an algebraic-geometric object built from the interplay between matroid theory and the theory of ideals generated by specific combinatorial or geometric configurations, notably generalizing classic uniform matroids. This construction extends the notion of star configurations and matroid Stanley–Reisner ideals, providing a framework in which the rich combinatorial structure of generalized uniform matroids manifests in the algebraic properties of their associated ideals. The theory connects commutative algebra, algebraic geometry, and matroid combinatorics, yielding sharp results on symbolic powers, resurgence numbers, and Cohen–Macaulayness.

1. Definition and Algebraic Construction

Let $T = k[x_1, \ldots, x_s]$ be a polynomial ring over a field $k$. Fix integers $1 \leq c \leq n \leq s$.

• The uniform matroid $U_{c,n}$ is the matroid whose ground set is a chosen $n$-element subset of $\{1, \ldots, s\}$, with independent sets the subsets of size at most $c$. Its circuits are exactly the $(c+1)$-element subsets.
• The $C$-matroidal ideal of $U_{c,n}$ is

$$I = \bigcap_{\substack{J \subset \{1, \ldots, n\} \ |J| = c}} (x_j : j \in J) \subset T,$$

i.e., $I$ is the intersection of all coordinate primes of height $c$ supported on those $n$ variables. Combinatorially, $I$ is the cover ideal $J(\Delta)$ of the uniform-matroid simplicial complex $\Delta$.

For any graded $k$-algebra $R$ and a collection of homogeneous forms $(f_1, \ldots, f_s) \in R$, if any $c+1$ of the $f_i$ form a regular sequence, the specialization map $\phi: T \to R,~x_i \mapsto f_i$ defines a new ideal $I_* = \phi(I) \subset R$—this is termed a generalized uniform matroidal configuration. When $s=n$, this recovers the star configuration of hypersurfaces (Hu, 16 Nov 2025, Geramita et al., 2015).

2. Matroidal Ideals and Symbolic Powers

The minimal generators of the defining ideal are indexed by the circuits of the matroid. For the monomial ideal $I$ arising from a uniform matroid, the symbolic power $I^{(m)}$ admits a combinatorial description:

$$I^{(m)} = \bigcap_{\substack{J \subset \{1,\ldots,n\} \ |J|=c}} (x_j : j \in J)^m$$

Specializing to a generalized uniform matroidal configuration $I_*$, the symbolic power $I_*^{(m)}$ is the intersection of $m$th powers of the defining ideals of the associated complete intersections. These symbolic powers are always Cohen–Macaulay, and their minimal generators correspond to monomials that vanish to order at least $m$ on each component (Geramita et al., 2015).

3. Resurgence and Waldschmidt Constants

The resurgence number $\rho(I)$ of a generalized uniform matroidal configuration quantifies the failure of containment of symbolic powers in ordinary powers:

$$\rho(I) = \sup \left\{ \frac{m}{r} : I^{(m)} \not\subseteq I^r \right\}$$

For the $C$-matroidal ideal of $U_{c,n}$, an explicit formula holds:

$\rho(I) = \frac{c(n-c+1)}{n}$

Moreover, strict containment $I^{(m)}\not\subseteq I^r$ occurs exactly when

$\frac{m}{r} > \frac{n}{c(n-c+1)}$

This result extends directly to their specializations $I_*$, including star configurations and more general hypersurface configurations (Hu, 16 Nov 2025, Geramita et al., 2015).

The Waldschmidt constant $\widehat\alpha(I)$, giving the asymptotic initial degree of symbolic powers, also admits a closed formula in the uniform case:

$\widehat\alpha(I_{U_{r,n}}) = \frac{r+1}{n}$

4. Connection to Stanley–Reisner Ideals and Projective Geometry

The Stanley–Reisner ideal of $U_{r,n}$ is generated by all squarefree monomials of degree $r+1$. After specialization by $y_i \mapsto f_i$, one recovers the ideal of a union of codimension-$c$ complete intersections (in the projective or affine setting, depending on the nature of the $f_i$). The corresponding scheme—the uniform-matroid configuration of type $(r,n,c)$—is the union

$$V_{r,n,c} = \bigcup_{1 \leq i_1 < \cdots < i_c \leq n} X_{i_1, \ldots, i_c}$$

with $X_{i_1, \ldots, i_c}$ the codimension-$c$ intersection of hypersurfaces $f_{i_j}=0$. The ideal is generated by all products of $n-c+1$ of the $f_i$, and flat specialization preserves Hilbert function and free resolution structure (Geramita et al., 2015).

5. Syzygies, Cohen–Macaulayness, and Homological Properties

The ideals of generalized uniform matroidal configurations possess linear minimal free resolutions, with Betti numbers governed by the combinatorics of the underlying matroid:

$$0 \longrightarrow \bigoplus R(-d_{n-r})^{\beta_{n-r-1}} \longrightarrow \cdots \longrightarrow R(-d_{1})^{\beta_1} \longrightarrow R(-d_0)^{\beta_0} \longrightarrow I \longrightarrow 0,$$

where $\beta_i = \binom{r+i}{i} \binom{n}{r+1+i}$ and all syzygies are linear. All symbolic powers are Cohen–Macaulay, a consequence of the Alexander duality of matroid complexes and the Eagon–Reiner theorem: this extends to all specializations as long as the $f_i$ remain sufficiently generic (Geramita et al., 2015).

6. Strict Containment Criteria and Peaked Simplicial Complexes

A sharp criterion for the containment $I^{(m)} \subseteq I^r$ is given in terms of degrees and combinatorics:

$$I^{(m)} \subseteq I^r \Longleftrightarrow r \geq \frac{\frac{m}{c}(n-c) + m}{n-c+1}$$

This formula determines all pairs $(m,r)$ for which symbolic powers are contained in ordinary powers. The theory further generalizes to peaked simplicial complexes—a class encompassing uniform-matroid configurations as a special case—and provides upper bounds for the resurgence in those settings (Hu, 16 Nov 2025).

7. Illustrative Examples and Special Cases

Prominent examples include:

• Star configurations of points/hyperplanes in projective space (the case $n=s$), with $\rho(I) = 2n/s$ for points on a projective line, matching classical results.
• Hypersurface star-configurations in affine or projective $c$-space, with $\rho(I) = c(s-c+1)/s$; this matches previous results on star and hypersurface configurations.

Generalized uniform matroidal configurations thus encompass a broad class of highly structured ideals, linking matroid combinatorics with the algebraic and geometric properties of their associated schemes. This theory clarifies the interplay between containment problems in algebra and the combinatorics of generalized uniform matroids (Hu, 16 Nov 2025, Geramita et al., 2015).