Abstract
Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its nonsmooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen–Macaulay of codimension 3. If the matroid is 3connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions.
Introduction
Feynman diagrams
A basic problem in highenergy physics is to understand the scattering of particles. The basic tool for theoretical predictions is the Feynman diagram with underlying Feynman graph \(G=(V,E)\). The scattering data correspond to Feynman integrals, computed in the positive orthant of the projective space labeled by the internal edges of the Feynman graph. The integrand is the square root of a rational function in the edge variables \(x_e\), \(e\in E\), that depends parametrically on the masses and moments of the involved particles (see [10]).
The convergence of a Feynman integral is determined by the structure of the denominator of this rational function, which always involves a power of the square root of the Symanzik polynomial \(\sum _{T\in \mathcal {T}_G}\prod _{e\not \in T}x_e\) of G where \(\mathcal {T}_G\) denotes the set of spanning trees of G. The remaining factor of the denominator, appearing for graphs with edge number less than twice the loop number, is a power of the square root of the second Symanzik polynomial obtained by summing over 2forests and involves masses and moments. Symanzik polynomials can factor, and the singularities and intersections of the individual components determine the behavior of the Feynman integrals.
Until about a decade ago, all explicitly computed integrals were built from multiple Riemann zeta values and polylogarithms; for example, Broadhurst and Kreimer display a large body of such computations in [8]. In fact, Kontsevich at some point speculated that Symanzik polynomials, or equivalently their cousins the Kirchhoff polynomials
be mixed Tate; this would imply the relation to multiple zeta values. However, Belkale and Brosnan [4] proved that the collection of Kirchhoff polynomials is a rather complicated class of singularities: their hypersurface complements generate the ring of all geometric motives. This does not exactly rule out that Feynman integrals are in some way wellbehaved, but makes it rather less likely, and explicit counterexamples to Kontsevich’s conjecture were subsequently worked out by Doryn [15] as well as by Brown and Schnetz [11]. On the other hand, these examples make the study of these singularities, and especially any kind of uniformity results, that much more interesting.
The influential paper [6] of Bloch, Esnault and Kreimer generated a significant amount of work from the point of view of complex geometry: we refer to the book [23] of Marcolli for exposition, as well as [10, 12, 15]. Varying ideas of Connes and Kreimer on renormalization that view Feynman integrals as specializations of the Tutte polynomial, Aluffi and Marcolli formulate in [1, 2] parametric Feynman integrals as periods, leading to motivic studies on cohomology. On the explicit side, there is a large body of publications in which specific graphs and their polynomials and Feynman integrals are discussed. But, as Brown writes in [9], while a diversity of techniques is used to study Feynman diagrams, “each new loop order involves mathematical objects which are an order of magnitude more complex than the last, [...] the unavoidable fact is that arbitrary integrals remain out of reach as ever.”
The present article can be seen as the first step towards a search for uniform properties in this zoo of singularities. We view it as a stepping stone for further studies of invariants such as log canonical threshold, logarithmic differential forms and embedded resolution of singularities.
Configuration polynomials
The main idea of Belkale and Brosnan is to move the burden of proof into the more general realm of polynomials and constructible sets derived from matroids rather than graphs, and then to reduce to known facts about such polynomials. The article [6] casts Kirchhoff and Symanzik polynomials as very special instances of configuration polynomials; this idea was further developed by Patterson in [27]. We consider this as a more natural setting since notions such as duality and quotients behave well for configuration polynomials as a whole, but these operations do not preserve the subfamily of matroids derived from graphs. In particular, we can focus exclusively on Kirchhoff/configuration polynomials, since the Symanzik polynomial of G appears as the configuration polynomial of the dual configuration induced by the incidence matrix of G.
The configuration polynomial does not depend on a matroid itself but on a configuration, that is, on a (linear) realization of a matroid over a field \(\mathbb {K}\). The same matroid can admit different realizations, which, in turn, give rise to different configuration polynomials (see Example 5.3). The matroid (basis) polynomial is a competing object, which is assigned to any, even nonrealizable, matroid. It has proven useful for combinatorial applications (see [3, 28]). For graphs and, more generally, regular matroids, all configuration polynomials essentially agree with the matroid polynomial. In general, however, configuration polynomials differ significantly from matroid polynomials, as documented in Example 5.2.
Configuration polynomials have a geometric feature that matroid polynomials lack: generalizing Kirchhoff’s matrixtree theorem, the configuration polynomial arises as the determinant of a symmetric bilinear configuration form with linear polynomial coefficients. As a consequence, the corresponding configuration hypersurface maps naturally to the generic symmetric determinantal variety. In the present article, we establish further uniform, geometric properties of configuration polynomials, which we observe do not hold for matroid polynomials in general.
Summary of results
Some indication of what is to come can be gleaned from the following note by Marcolli in [23, p. 71]: “graph hypersurfaces tend to have singularity loci of small codimension.”
Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\) of rank \({{\,\mathrm{rk}\,}}\mathsf {M}=\dim W\) on a set E (see Definition 2.14). Fix coordinates \(x_E=(x_e)_{e\in E}\). There is an associated symmetric configuration (bilinear) form \(Q_W\) with linear homogeneous coefficients (see Definition 3.20). Its determinant is the configuration polynomial (see Definition 3.2 and Lemma 3.23)
where \(\mathcal {B}_\mathsf {M}\) denotes the set of bases of \(\mathsf {M}\) and the coefficients \(c_{W,B}\in \mathbb {K}^*\) depend of the realization W. The configuration hypersurface defined by \(\psi _W\) is the scheme
It can be seen as the first degeneracy scheme of \(Q_W\) (see Definition 4.9). The second degeneracy scheme \(\Delta _W\subseteq \mathbb {K}^E\) of \(Q_W\), defined by the submaximal minors of \(Q_W\), is a subscheme of the Jacobian scheme \(\Sigma _W\subseteq \mathbb {K}^E\) of \(X_W\), defined by \(\psi _W\) and its partial derivatives (see Lemma 4.12). The latter defines the nonsmooth locus of \(X_W\) over \(\mathbb {K}\), which is the singular locus of \(X_W\) if \(\mathbb {K}\) is perfect (see Remark 4.10). Patterson showed \(\Sigma _W\) and \(\Delta _W\) have the same underlying reduced scheme (see Theorem 4.17), that is,
We give a simple proof of this fact. He mentions that he does not know the reduced scheme structure (see [27, p. 696]). We show that \(\Sigma _W\) is typically not reduced (see Example 5.1), whereas \(\Delta _W\) often is. Our main results from Theorems 4.16, 4.25, 4.36 and 4.37 can be summarized as follows.
Main Theorem Let \(\mathsf {M}\) be a matroid on the set E with a linear realization \(W\subseteq \mathbb {K}^E\) over a field \(\mathbb {K}\). Then the configuration hypersurface \(X_W\) is reduced and generically smooth over \(\mathbb {K}\). Moreover, the second degeneracy scheme \(\Delta _W\) is also reduced and agrees with \(\Sigma _W^\text {red}\), the nonsmooth locus of \(X_W\) over \(\mathbb {K}\). Unless \(\mathbb {K}\) has characteristic 2, the Jacobian scheme \(\Sigma _W\) is generically reduced.
Suppose now that \(\mathsf {M}\) is connected. Then \(X_W\) is integral unless \(\mathsf {M}\) has rank zero. Suppose in addition that the rank of \(\mathsf {M}\) is at least 2. Then \(\Delta _W\) is Cohen–Macaulay of codimension 3 in \(\mathbb {K}^E\). If, moreover, \(\mathsf {M}\) is 3connected, then \(\Delta _W\) is integral. \(\square \)
Note that \(X_W=\emptyset \) if \({{\,\mathrm{rk}\,}}\mathsf {M}=0\) and \(\Sigma _W=\emptyset =\Delta _W\) if \({{\,\mathrm{rk}\,}}\mathsf {M}\le 1\) (see Remarks 3.5 and 4.13.(a)). It suffices to require the connectedness hypotheses after deleting all loops (see Remark 4.11). If \(\mathsf {M}\) is disconnected even after deleting all loops, then \(\Sigma _W\) and hence \(\Delta _W\) has codimension 2 in \(\mathbb {K}^E\) (see Proposition 4.16).
While our main objective is to establish the results above, along the way we continue the systematic study of configuration polynomials in the spirit of [6, 27]. For instance, we describe the behavior of configuration polynomials with respect to connectedness, duality, deletion/contraction and 2separations (see Propositions 3.8, 3.10, 3.12 and 3.27). Patterson showed that the second Symanzik polynomial associated with a Feynman graph is, in fact, a configuration polynomial. More precisely, we explain that its dual, the second Kirchhoff polynomial, is associated with the quotient of the graph configuration by the momentum parameters (see Proposition 3.19). In this way, Patterson’s result becomes a special case of a formula for configuration polynomials of elementary quotients (see Proposition 3.14).
Outline of the proof
The proof of the Main Theorem intertwines methods from matroid theory, commutative algebra and algebraic geometry. In order to keep our arguments selfcontained and accessible, we recall preliminaries from each of these subjects and give detailed proofs (see §2.1, §2.3 and §4.1). One easily reduces the claims to the case where \(\mathsf {M}\) is connected (see Proposition 3.8 and Theorem 4.36).
An important commutative algebra ingredient is a result of Kutz (see [22]): the grade of an ideal of submaximal minors of a symmetric matrix cannot exceed 3, and equality forces the ideal to be perfect. Kutz’ result applies to the defining ideal of \(\Delta _W\). The codimension of \(\Delta _W\) in \(\mathbb {K}^E\) is therefore bounded by 3 and \(\Delta _W\) is Cohen–Macaulay in case of equality (see Proposition 4.19). In this case, \(\Delta _W\) is puredimensional, and hence, it is reduced if it is generically reduced (see Lemma 4.4).
On the matroid side our approach makes use of handles (see Definition 2.3), which are called ears in case of graphic matroids. A handle decomposition builds up any connected matroid from a circuit by successively attaching handles (see Definition 2.6). Conversely, this yields for any connected matroid which is not a circuit a nondisconnective handle which leaves the matroid connected when deleted (see Definition 2.3). This allows one to prove statements on connected matroids by induction.
We describe the effect of deletion and contraction of a handle H to the configuration polynomial (see Corollary 3.13). In case the Jacobian scheme \(\Sigma _{W{\setminus } H}\) associated with the deletion \(\mathsf {M}{\setminus } H\) has codimension 3 we prove the same for \(\Sigma _W\) (see Lemma 4.22). Applied to a nondisconnective H it follows with Patterson’s result that \(\Delta _W\) reaches the dimension bound and is thus Cohen–Macaulay of codimension 3 (see Theorem 4.25). We further identify three (more or less explicit) types of generic points with respect to a nondisconnective handle (see Corollary 4.26).
In case \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\), generic reducedness of \(\Sigma _W\) implies (generic) reducedness of \(\Delta _W\). The schemes \(\Sigma _W\) and \(\Delta _W\) show similar behavior with respect to deletion and contraction (see Lemmas 4.29 and 4.31). As a consequence, generic reducedness can be proved along the same lines (see Lemma 4.35). In both cases, we have to show reducedness at all (the same) generic points. In what follows, we restrict ourselves to \(\Delta _W\). Our proof proceeds by induction on the cardinality \({\left E\right }\) of the underlying set E of the matroid \(\mathsf {M}\).
Unless \(\mathsf {M}\) a circuit, the handle decomposition guarantees the existence of a nondisconnective handle H. In case \(H={\left\{ h\right\} }\) has size 1, the scheme \(\Delta _{W{\setminus } h}\) associated with the deletion \(\mathsf {M}{\setminus } h\) is the intersection of \(\Delta _W\) with the divisor \(x_e\) (see Lemma 4.29). This serves to recover generic reducedness of \(\Delta _W\) from \(\Delta _{W{\setminus } h}\) (see Lemma 4.30). The same argument works if H does not arise from a handle decomposition.
This leads us to consider nondisconnective handles independently of a handle decomposition. They turn out to be special instances of maximal handles which form the handle partition of the matroid (see Lemma 2.4). As a purely matroidtheoretic ingredient, we show that the number of nondisconnective handles is strictly increasing when adding handles (see Proposition 2.12). For handle decompositions of length 2, a distinguished role is played by the prism matroid (see Example 2.7). Its handle partition consists of 3 nondisconnective handles of size 2 (see Lemmas 2.10 and 2.25). Here an explicit calculation shows that \(\Delta _W\) is reduced in the torus \((\mathbb {K}^*)^6\) (see Lemma 4.28). The corresponding result for \(\Sigma _W\) holds only if \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\).
Suppose now that \(\mathsf {M}\) is not a circuit and has no nondisconnective handles of size 1. Then \(\mathsf {M}{\setminus } e\) might be disconnected for all \(e\in E\) and does not qualify for an inductive step. In this case, we aim instead for contracting W by a suitable subset \(G\subsetneq E\) which keeps \(\mathsf {M}\) connected. In the partial torus \(\mathbb {K}^F\times (\mathbb {K}^*)^G\) where \(F:=E{\setminus } G\), the scheme \(\Delta _{W/G}\) associated with the contraction \(\mathsf {M}/G\) relates to the normal cone of \(\Delta _W\) along the coordinate subspace \(V(x_F)\) where \(x_F=(x_f)_{f\in F}\) (see Lemma 4.31). To induce generic reducedness from \(\Delta _{W/G}\) to \(\Delta _W\), we pass through a deformation to the normal cone, which is our main ingredient from algebraic geometry. The role of \(x_h\) above is then played by the deformation parameter t.
In algebraic terms, this deformation is represented by the Rees algebra \({{\,\mathrm{Rees}\,}}_IR\) with respect to an ideal \(I\unlhd R\), and the normal cone by the associated graded ring \({{\,\mathrm{gr}\,}}_IR\) (see Definition 4.6). Passing through \({{\,\mathrm{Rees}\,}}_IR\), we recover generic reducedness of R along V(I) from generic reducedness of \({{\,\mathrm{gr}\,}}_IR\) (see Definition 4.3 and Lemma 4.7). By assumption on \(\mathsf {M}\), there are at least 3 more elements in E than maximal handles (see Proposition 2.12), and \(\mathsf {M}\) is the prism matroid in case of equality. Based on a strict inequality, we use a codimension argument to construct a suitable partition \(E=F\sqcup G\) for which all generic points of \(\Delta _W\) are along \(V(x_F)\) (see Lemma 4.34). This yields generic reducedness of \(\Delta _W\) in this case (see Lemma 4.32). A slight modification of the approach also covers the generic points outside the torus \((\mathbb {K}^*)^6\) if \(\mathsf {M}\) is the prism matroid. The case where \(\mathsf {M}\) is a circuit is reduced to that where \(\mathsf {M}\) is a triangle by successively contracting an element of E (see Lemma 4.33). In this base case \(\Delta _W\) is a reduced point, but \(\Sigma _W\) is reduced only if \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\) (see Example 4.14).
Finally, suppose that \(\mathsf {M}\) is a 3connected matroid. Here we prove that \(\Delta _W\) is irreducible and hence integral, which implies that \(\Sigma \) is irreducible (see Theorem 4.37). We first observe that handles of (co)size at least 2 are 2separations (see Lemma 2.4.(e)). It follows that the handle decomposition consists entirely of nondisconnective 1handles (see Proposition 2.5) and that all generic points of \(\Delta _W\) lie in the torus \((\mathbb {K}^*)^E\) (see Corollary 4.27). We show that the number of generic points is bounded by that of \(\Delta _{W{\setminus } e}\) for all \(e\in E\) (see Lemma 4.30). Duality switches deletion and contraction and identifies generic points of \(\Delta _W\) and \(\Delta _{W^\perp }\) (see Corollary 4.18). Using Tutte’s wheelsandwhirls theorem, the irreducibility of \(\Delta _W\) can therefore be reduced to the cases where \(\mathsf {M}\) is a wheel \(\mathsf {W}_n\) or a whirl \(\mathsf {W}^n\) for some \(n\ge 3\) (see Example 2.26 and Lemma 4.38). For fixed n, we show that the schemes \(X_W\), \(\Sigma _W\) and \(\Delta _W\) are all isomorphic for all realizations W of \(\mathsf {W}_n\) and \(\mathsf {W}^n\) (see Proposition 4.40). An induction on n with an explicit study of the base cases \(n\le 4\) finishes the proof (see Corollary 4.41 and Lemma 4.43).
Matroids and configurations
Our algebraic objects of interest are associated with a realization of a matroid. In this section, we prepare the path for an inductive approach driven by the underlying matroid structure. Our main tool is the handle decomposition, a matroid version of the ear decomposition of graphs.
Matroid basics
In this subsection, we review the relevant basics of matroid theory using Oxley’s book (see [26]) as a comprehensive reference.
Denote by \({{\,\mathrm{Min}\,}}\mathcal {P}\) and \({{\,\mathrm{Max}\,}}\mathcal {P}\) the set of minima and maxima of a poset \(\mathcal {P}\). Let \(\mathsf {M}\) be a matroid on a set \(E=:E_\mathsf {M}\). We use this font throughout to denote matroids. With \(2^E\) partially ordered by inclusion, \(\mathsf {M}\) can be defined by a monotone submodular rank function (see [26, Cor. 1.3.4])
with \({{\,\mathrm{rk}\,}}(S)\le {\left S\right }\) for any subset \(S\subseteq E\). The rank of \(\mathsf {M}\) is then
Alternatively, it can be defined in terms of each of the following collections of subsets of E (see [26, Prop. 1.3.5, p. 28]):

independent sets \(\mathcal {I}_\mathsf {M}={\left\{ I\subseteq E\;\big \;{\left I\right }={{\,\mathrm{rk}\,}}_\mathsf {M}(I)\right\} }\subseteq 2^E\),

bases \(\mathcal {B}_\mathsf {M}={{\,\mathrm{Max}\,}}\mathcal {I}_\mathsf {M}={\left\{ B\subseteq E\;\big \;{\left B\right }={{\,\mathrm{rk}\,}}_\mathsf {M}(B)={{\,\mathrm{rk}\,}}\mathsf {M}\right\} }\subseteq 2^E\),

circuits \(\mathcal {C}_\mathsf {M}={{\,\mathrm{Min}\,}}(2^E{\setminus }\mathcal {I}_\mathsf {M})\subseteq 2^E\),

flats \(\mathcal {L}_\mathsf {M}={\left\{ F\subseteq E \;\big \;\forall e\in E{\setminus } F:{{\,\mathrm{rk}\,}}_\mathsf {M}(F\cup {\left\{ e\right\} })>{{\,\mathrm{rk}\,}}_\mathsf {M}(F)\right\} }\).
For instance (see [26, Lem. 1.3.3]), for any subset \(S\subseteq E\),
The closure operator of \(\mathsf {M}\) is defined by (see [26, Lem. 1.4.2])
The following matroid plays a special role in the proof of our main result.
Definition 2.1
(Prism matroid). The prism matroid has underlying set E with \({\left E\right }=6\) and circuits
The name comes from the observation that its independent sets \(\mathcal {I}_{\mathsf {M}}\) are the affinely independent subsets of the vertices of the triangular prism (see Fig. 1).
The elements of \(E{\setminus }\bigcup \mathcal {B}_\mathsf {M}\) and \(\bigcap \mathcal {B}_\mathsf {M}\) are called loops and coloops in \(\mathsf {M}\), respectively. A matroid is free if \(E\in \mathcal {B}_\mathsf {M}\), that is, every \(e\in E\) is a coloop in \(\mathsf {M}\). By a kcircuit in \(\mathsf {M}\) we mean a circuit \(C\in \mathcal {C}_\mathsf {M}\) with \({\left C\right }=k\) elements, 3circuits are called triangles.
The circuits in \(\mathsf {M}\) give rise to an equivalence relation on E by declaring \(e,f\in E\) equivalent if \(e=f\) or \(e,f\in C\) for some \(C\in \mathcal {C}_\mathsf {M}\) (see [26, Prop. 4.1.2]). The corresponding equivalence classes are the connected components of \(\mathsf {M}\). If there is at most one such a component, then \(\mathsf {M}\) is said to be connected. The connectivity function of \(\mathsf {M}\) is defined by
For \(k\ge 1\), a subset \(S\subseteq E\), or the partition \(E=S\sqcup (E{\setminus } S)\), is called a kseparation of \(\mathsf {M}\) if
It is called exact if the latter is an equality. The connectivity \(\lambda (\mathsf {M})\) of \(\mathsf {M}\) is the minimal k for which there is a kseparation of \(\mathsf {M}\), or \(\lambda (\mathsf {M})=\infty \) if no such exists. The matroid \(\mathsf {M}\) is said to be kconnected if \(\lambda (\mathsf {M})\ge k\). Connectedness is the special case \(k=2\).
We now review some standard constructions of new matroids from old. Their geometric significance is explained in §2.3.
The direct sum \(\mathsf {M}_1\oplus \mathsf {M}_2\) of matroids \(\mathsf {M}_1\) and \(\mathsf {M}_2\) is the matroid on \(E_{\mathsf {M}_1}\sqcup E_{\mathsf {M}_2}\) with independent sets
The sum is proper if \(E_{\mathsf {M}_1}\ne \emptyset \ne E_{\mathsf {M}_2}\). Connectedness means that a matroid is not a proper direct sum (see [26, Prop. 4.2.7]). In particular, any (co)loop is a connected component.
Let \(F\subseteq E\) be any subset. Then the restriction matroid \(\mathsf {M}\vert _F\) is the matroid on F with independent sets and bases (see [26, 3.1.12, 3.1.14])
Its set of circuits is (see [26, 3.1.13])
By definition, \({{\,\mathrm{rk}\,}}_{\mathsf {M}\vert _F}={{\,\mathrm{rk}\,}}_\mathsf {M}\vert _{2^F}\), so we may omit the index without ambiguity. Thinking of restriction to \(E{\setminus } F\) as an operation that deletes elements in F from E, one defines the deletion matroid
The contraction matroid \(\mathsf {M}/F\) is the matroid on \(E{\setminus } F\) with independent sets and bases (see [26, Prop. 3.1.7, Cor. 3.1.8])
Its circuits are the minimal nonempty sets \(C{\setminus } F\) where \(C\in \mathcal {C}_{\mathsf {M}}\) (see [26, Prop. 3.1.10]), that is,
In §2.3, E will be a basis and \(E^\vee \) the corresponding dual basis. We often identify \(E=E^\vee \) by the bijection
The complement of a subset \(S\subseteq E\) corresponds to
The dual matroid \(\mathsf {M}^\perp \) is the matroid on \(E^\vee \) with bases
In particular, we have (see [26, 2.1.8])
Connectivity is invariant under dualizing (see [26, Cor. 8.1.5]),
We use \(\nu ^{1}\) in place of (2.8) for \(\mathsf {M}^\perp \), so that \(S^{\perp \perp }=S\). For subsets \(F\subseteq E\) and \(G\subseteq E^\vee \), one can identify (see [26, 3.1.1])
Various matroid data of \(\mathsf {M}^\perp \) is also considered as codata of \(\mathsf {M}\). A triad of \(\mathsf {M}\) is a 3cocircuit of \(\mathsf {M}\), that is, a triangle of \(\mathsf {M}^\perp \).
Example 2.2
(Uniform matroids). The uniform matroid \(\mathsf {U}_{r,n}\) of rank \(r\ge 0\) on a set E of size \({\left E\right }=n\) has bases
For \(r=n\) it is the free matroid of rank r. It is connected if and only if \(0<r<n\). By definition, \(\mathsf {U}_{r,n}^\perp =\mathsf {U}_{nr,n}\) for all \(0\le r\le n\).
Informally, we refer to a matroid \(\mathsf {M}\) on E for which \(E\in \mathcal {C}_\mathsf {M}\), and hence, \(\mathcal {C}_\mathsf {M}={\left\{ E\right\} }\), as a circuit, and as a triangle if \({\left E\right }=3\). It is easily seen that such a matroid is \(\mathsf {U}_{n1,n}\) where \(n={\left E\right }\), and that \(\lambda (\mathsf {U}_{n1,n})=2\).
Handle decomposition
In this subsection, we investigate handles as building blocks of connected matroids.
Definition 2.3
(Handles). Let \(\mathsf {M}\) be a matroid. A subset \(\emptyset \ne H\subseteq E\) is a handle in \(\mathsf {M}\) if \(C\cap H\ne \emptyset \) implies \(H\subseteq C\) for all \(C\in \mathcal {C}_\mathsf {M}\). Write \(\mathcal {H}_\mathsf {M}\) for the set of handles in \(\mathsf {M}\), ordered by inclusion. A subhandle of \(H\in \mathcal {H}_\mathsf {M}\) is a subset \(\emptyset \ne H'\subseteq H\). We call \(H\in \mathcal {H}_\mathsf {M}\)

proper if \(H\ne E\),

maximal if \(H\in {{\,\mathrm{Max}\,}}\mathcal {H}_\mathsf {M}\),

a khandle if \({\left H\right }=k\),

disconnective if \(\mathsf {M}{\setminus } H\) is disconnected and

separating if \(\min {\left\{ {\left H\right },{\left E{\setminus } H\right }\right\} }\ge 2\).
Singletons \({\left\{ e\right\} }\) and subhandles are handles. If \(\bigcup \mathcal {C}_\mathsf {M}\ne E\), then \(E{\setminus }\bigcup \mathcal {C}_\mathsf {M}\in {{\,\mathrm{Max}\,}}\mathcal {H}_\mathsf {M}\) and is a union of coloops. The maximal handles in \(\bigcup \mathcal {C}_\mathsf {M}\) are the minimal nonempty intersections of all subsets of \(\mathcal {C}_\mathsf {M}\). Together they form the handle partition of E
which refines the partition of \(\bigcup \mathcal {C}_\mathsf {M}\) into connected components. In particular, each circuit is a disjoint union of maximal handles. For any subset \(F\subseteq E\), (2.5) yields an inclusion
Lemma 2.4
(Handle basics). Let \(\mathsf {M}\) be a matroid and \(H\in \mathcal {H}_\mathsf {M}\).

(a)
If \(H=E\), then \(\mathsf {M}=\mathsf {U}_{r,n}\) where \(n={\left E\right }\ge 1\) and \(r\in {\left\{ n1,n\right\} }\) (see Example 2.2). In the latter case, \({\left E\right }=1\) if \(\mathsf {M}\) is connected.

(b)
Either \(H\in \mathcal {I}_\mathsf {M}\) or \(H\in \mathcal {C}_\mathsf {M}\). In the latter case, H is maximal and a connected component of \(\mathsf {M}\). In particular, if \(\mathsf {M}\) is connected and H is proper, then \(H\in \mathcal {I}_\mathsf {M}\), \(H\subsetneq C\) for some circuit \(C\in \mathcal {C}_\mathsf {M}\), and \(H\in \mathcal {C}_{\mathsf {M}/(E{\setminus } H)}\).

(c)
For any subhandle \(\emptyset \ne H'\subseteq H\), \(H{\setminus } H'\) consists of coloops in \(\mathsf {M}{\setminus } H'\). In particular, nondisconnective handles are maximal.

(d)
If \(H\not \in \mathcal {C}_\mathsf {M}\), then there is a bijection
$$\begin{aligned} \mathcal {C}_\mathsf {M}\rightarrow \mathcal {C}_{\mathsf {M}/H},\quad C\mapsto C{\setminus } H. \end{aligned}$$If \(H\not \in {{\,\mathrm{Max}\,}}\mathcal {H}_\mathsf {M}\), then there is a bijection
$$\begin{aligned} {{\,\mathrm{Max}\,}}\mathcal {H}_\mathsf {M}\rightarrow {{\,\mathrm{Max}\,}}\mathcal {H}_{\mathsf {M}/H},\quad H'\mapsto H'{\setminus } H, \end{aligned}$$which identifies nondisconnective handles. In this case, the connected components of \(\mathsf {M}\) which are not contained in \(H{\setminus }\bigcup \mathcal {C}_\mathsf {M}\) correspond to the connected components of \(\mathsf {M}/H\).

(e)
Suppose that \(\mathsf {M}\) is connected and H is proper. Then
$$\begin{aligned} {{\,\mathrm{rk}\,}}(\mathsf {M}/H)={{\,\mathrm{rk}\,}}\mathsf {M}{\left H\right },\quad \lambda _\mathsf {M}(H)=1. \end{aligned}$$In particular, if H is separating, then H is a 2separation of \(\mathsf {M}\).
Proof

(a)
Suppose that \(H=E\). Then \(\mathcal {C}_\mathsf {M}\subseteq {\left\{ E\right\} }\) and \(\mathsf {M}=\mathsf {U}_{n1,n}\) in case of equality. Otherwise, \(\mathcal {C}_\mathsf {M}=\emptyset \) implies \(\mathcal {B}_\mathsf {M}={\left\{ E\right\} }\) and \(\mathsf {M}=\mathsf {U}_{n,n}\) (see [26, Prop. 1.1.6]).

(b)
Suppose that \(H\not \in \mathcal {I}_\mathsf {M}\). Then there is a circuit \(H\supseteq C\in \mathcal {C}_\mathsf {M}\). By definition of handle and incomparability of circuits, \(H=C{\setminus }(E\setminus H)\in \mathcal {C}_{\mathsf {M}/(E{\setminus } H)}\) (see (2.7)) and \(H=C\) is disjoint from all other circuits and hence a connected component of \(\mathsf {M}\).

(c)
Suppose that \(h\in H{\setminus } H'\) is not a coloop in \(\mathsf {M}{\setminus } H'\). Then \(h\in C\cap H\) for some \(C\in \mathcal {C}_{\mathsf {M}{\setminus } H'}\subseteq \mathcal {C}_\mathsf {M}\) (see (2.5)) and hence \(H'\subseteq H\subseteq C\) since H is a handle, a contradiction.

(d)
The first bijection follows from (2.7) with \(F=H\). The remaining claims follow from the discussion preceding the lemma.

(e)
Part (b) yields the first equality (see [26, Prop. 3.1.6]) along with a circuit \(H\ne C\in \mathcal {C}_\mathsf {M}\). Pick a basis \(B\in \mathcal {B}_{\mathsf {M}{\setminus } H}\). Clearly \(S:=B\sqcup H\) spans \(\mathsf {M}\). For any \(h\in H\), we check that \(S{\setminus }{\left\{ h\right\} }\in \mathcal {I}_\mathsf {M}\). Otherwise, there is a circuit \(S{\setminus }{\left\{ h\right\} }\supseteq C\in \mathcal {C}_\mathsf {M}\). Since \(C\not \subseteq B\) and by definition of handle, we have \(H\cap C\ne \emptyset \) and hence \(h\in H\subseteq C\), a contradiction. It follows that \({{\,\mathrm{rk}\,}}\mathsf {M}={\left S\right }1={{\,\mathrm{rk}\,}}(\mathsf {M}{\setminus } H)+{\left H\right }1\) and hence the second equality. \(\square \)
Proposition 2.5
(Handles in 3connected matroids). Let \(\mathsf {M}\) be a 3connected matroid on E with \({\left E\right }>3\). Then all its handles are nondisconnective 1handles.
Proof
Let \(H\in \mathcal {H}_\mathsf {M}\) be any handle. By Lemma 2.4.(a), H must be proper. By Lemma 2.4.(e), H is not separating, that is, \({\left H\right }=1\) or \({\left E{\setminus } H\right }=1\). In the latter case, \(\mathsf {M}\) is a circuit by Lemma 2.4.(b) and hence not 3connected (see Example 2.2). So H is a 1handle.
Suppose that H is disconnective. Consider the deletion \(\mathsf {M}':=\mathsf {M}{\setminus } H\) on the set \(E':=E{\setminus } H\). Pick a connected component X of \(\mathsf {M}'\) of minimal size \({\left X\right }<{\left E'\right }\). Since \(H\ne \emptyset \) and \({\left E\right }>3\), both \(X\cup H\) and its complement \(E{\setminus } (X\cup H)=E'{\setminus } X\) have at least 2 elements. Since X is a connected component of \(\mathsf {M}'\) and by Lemma 2.4.(e),
Since \({{\,\mathrm{rk}\,}}(X\cup H)\le {{\,\mathrm{rk}\,}}(X)+{\left H\right }={{\,\mathrm{rk}\,}}(X)+1\), it follows that
Whence \(X\cup H\) is a 2separation of \(\mathsf {M}\), a contradiction. \(\square \)
The following notion is the basis for our inductive approach to connected matroids.
Definition 2.6
(Handle decompositions). Let \(\mathsf {M}\) be a connected matroid. A handle decomposition of length k of \(\mathsf {M}\) is a filtration
such that \(\mathsf {M}\vert _{F_i}\) is connected and \(H_i:=F_i{\setminus } F_{i1}\in \mathcal {H}_{\mathsf {M}\vert _{F_i}}\) for \(i=2,\dots ,k\).
By Lemma 2.4.(b) and (2.5), a handle decomposition yields circuits
Conversely, it can be constructed from a suitable sequence of circuits.
Example 2.7
(Handle decomposition of the prism matroid). The prism matroid (see Example 2.1) has handle partition
A handle decomposition of length 2 is given by
Note that all handles are proper, maximal, separating 2handles.
Proposition 2.8
(Existence of handle decompositions). Let \(\mathsf {M}\) be a connected matroid and \(C_1\in \mathcal {C}_\mathsf {M}\). Then there is a handle decomposition of \(\mathsf {M}\) starting with \(F_1=C_1\).
Proof
There is a sequence of circuits \(C_1,\ldots ,C_k\in \mathcal {C}_\mathsf {M}\) which defines a filtration \(F_i:=\bigcup _{j\le i}C_j\) such that \(C_i\cap F_{i1}\ne \emptyset \) and \(C_i{\setminus } F_{i1}\in \mathcal {C}_{\mathsf {M}/F_{i1}}\) for \(i=2,\dots ,k\) (see [13]). The hypothesis \(C_i\cap F_{i1}\ne \emptyset \) implies that \(\mathsf {M}\vert _{F_i}\) is connected for \(i=1,\dots ,k\).
It remains to check that \(H_i=C_i{\setminus } F_{i1}\in \mathcal {H}_{\mathsf {M}\vert _{F_i}}\) for \(i=2,\dots ,k\). Since circuits are nonempty, \(\emptyset \ne H_i\subsetneq F_i\). Let \(C\in \mathcal {C}_{\mathsf {M}\vert _{F_i}}\) be a circuit such that \(e\in C\cap H_i\subseteq C\cap C_i\). Suppose by way of contradiction that \(H_i\not \subseteq C\). Then there exists some \(d\in C_i{\setminus }(C\cup F_{i1})\). By the strong circuit elimination axiom (see [26, Prop. 1.4.12]), there is a circuit \(C'\in \mathcal {C}_{\mathsf {M}\vert _{F_i}}\subseteq \mathcal {C}_\mathsf {M}\) (see (2.5)) for which \(d\in C'\subseteq (C\cup C_i) {\setminus }{\left\{ e\right\} }\). Then \(C'{\setminus } F_{i1}\subseteq C_i{\setminus } F_{i1}\in \mathcal {C}_{\mathsf {M}/F_{i1}}\) by assumption on \(C_i\). It follows that either \(C'\subseteq F_{i1}\) or \(C'{\setminus } F_{i1}=C_i{\setminus } F_{i1}\) (see (2.7)). The former is impossible because \(C'\ni d\not \in F_{i1}\), and the latter because \(C'\cup F_{i1}\not \ni e\in C_i\). \(\square \)
In the sequel, we develop a bound for the number of nondisconnective handles.
Lemma 2.9
(Nondisconnective handles). Let \(\mathsf {M}\) be a connected matroid. Suppose that \(H\in \mathcal {H}_\mathsf {M}\) and \(H'\in \mathcal {H}_{\mathsf {M}{\setminus } H}\) are nondisconnective with \(H\cup H'\ne E\). Then there is a nondisconnective handle \(H''\in \mathcal {H}_\mathsf {M}\) such that \(H''\subseteq H'\), with equality if \(H'\in \mathcal {H}_\mathsf {M}\).
Proof
By hypothesis, \(\mathsf {M}\) and \(\mathsf {M}{\setminus } H\) are connected and \(H\cup H'\ne E\) implies that both H and \(H'\) are proper handles. Then Lemma 2.4.(b) yields circuits \(C\in \mathcal {C}_\mathsf {M}\) and \(C'\in \mathcal {C}_{\mathsf {M}{\setminus } H}\subseteq \mathcal {C}_\mathsf {M}\) (see (2.5)) such that \(H\subsetneq C\) and \(H'\subsetneq C'\).
Suppose that \(C\subseteq H\cup H'\). Then the strong circuit elimination axiom (see [26, Prop. 1.4.12]) yields a circuit \(C''\in \mathcal {C}_{\mathsf {M}}\) for which \(C''\subseteq H\cup C'\), \(H'\not \subseteq C''\) and \(C''\not \subseteq H\cup H'\). Since \(C''\subsetneq C'\) contradicts incomparability of circuits, \(H\subsetneq C''\) since H is a handle and Lemma 2.4.(b) forbids equality.
Replacing C by \(C''\) if necessary, we may assume that \(H'\not \subseteq C\) and \(C\not \subseteq H\cup H'\). In particular, \(H'':=H'{\setminus } C\in \mathcal {H}_{\mathsf {M}{\setminus } H}\) and \(H''=H'\) if \(H'\in \mathcal {H}_\mathsf {M}\). Since \(\mathsf {M}{\setminus }(H\cup H')\) is connected by hypothesis, C witnesses the fact that H, \(C\cap H'\) and \(E{\setminus }(H\cup H')\) are in the same connected component of \(\mathsf {M}{\setminus } H''\) (see (2.5)). In other words, \(\mathsf {M}{\setminus } H''\) is connected. If \(H''\in \mathcal {H}_\mathsf {M}\) is a handle, then \(H''\) is therefore nondisconnective.
Otherwise, there is a circuit \(C''\in \mathcal {C}_\mathsf {M}\) such that \(\emptyset \ne C''\cap H''\ne H''\). In particular, \(H\subseteq C''\) since otherwise \(C''\cap H=\emptyset \) and \(C''\in \mathcal {C}_{\mathsf {M}{\setminus } H}\) (see (2.5)) which would contradict \(H''\in \mathcal {H}_{\mathsf {M}{\setminus } H}\). This means that \(C''\) connects H with \(C''\cap H''\). We may therefore replace \(H''\) by \(\emptyset \ne H''{\setminus } C''\subsetneq H''\) and iterate. Then \(H''\in \mathcal {H}_\mathsf {M}\) after finitely many steps. \(\square \)
Lemma 2.10
(Handle decomposition of length 2). Let \(\mathsf {M}\) be a connected matroid with a handle decomposition of length 2. Then \(\mathsf {M}\) has at least 3 (disjoint) nondisconnective handles. In case of equality, they form the handle partition of \(\mathsf {M}\).
Proof
Consider the circuits \(C':=C_1\in \mathcal {C}_\mathsf {M}\), \(C:=C_2\in \mathcal {C}_\mathsf {M}\) (see (2.12)), the nondisconnective handle \(H:=H_2\in \mathcal {H}_\mathsf {M}\) and the subsets \(\emptyset \ne H':=C'{\setminus } C\subseteq E\) and \(\emptyset \ne H'':=C\cap C'\subseteq E\). Then \(E=H\sqcup H'\sqcup H''\) and \(C'=H'\cup H''\) and \(C=H\cup H''\).
Let \(C''\in \mathcal {C}_\mathsf {M}\) be any circuit with \(C'\ne C''\ne C\). By incomparability of circuits, \(C''\not \subseteq C'\) and hence \(H\subseteq C''\) since H is a handle. By Lemma 2.4.(d), we may assume that \({\left H\right }=1\). Then \(H'\subseteq C''\) (see [26, §1.1, Exc. 5]). In particular, \(H'\in \mathcal {H}_\mathsf {M}\) is a third nondisconnective handle. If \(H\cup H'\subseteq C''\) is an equality, then also \(H''\in \mathcal {H}_\mathsf {M}\) is a nondisconnective handle and \(H\sqcup H'\sqcup H''\) is the handle decomposition.
Otherwise, \(C''\) witnesses the fact that H, \(H'\) and \(\emptyset \ne C''\cap H''\ne H''\) are in the same connected component of \(\mathsf {M}\vert _{C''}\) (see (2.5)). If \(H''{\setminus } C''\in \mathcal {H}_\mathsf {M}\) is a handle, then it is therefore nondisconnective. Otherwise, iterating yields a third nondisconnective handle \(H''{\setminus } C''\supseteq H'''\in \mathcal {H}_\mathsf {M}\). \(\square \)
Example 2.11
(Unexpected handles). Consider the matroid \(\mathsf {M}\) on \(E={\left\{ 1,\dots ,6\right\} }\) whose bases
index those sets of columns of the matrix
which form a basis of \(\mathbb {F}_3^4\) (see Remark 2.15). Its circuits and maximal handles are given by
In particular, \(\mathsf {M}\) is connected with a handle decomposition
of length 2. Here all 4 maximal handles are nondisconnective and the inequality in Lemma 2.10 is strict. This can happen because \(\mathsf {M}\) is not a graphic matroid (see Lemma 2.25).
Proposition 2.12
(Lower bound for nondisconnective handles). Let \(\mathsf {M}\) be a connected matroid with a handle decomposition of length \(k\ge 2\). Then \(\mathsf {M}\) has at least \(k+1\) (disjoint) nondisconnective handles.
Proof
We argue by induction on k. The base case \(k=2\) is covered by Lemma 2.10. Suppose now that \(k\ge 3\). By hypothesis (see Definition 2.6), \(H_k\in \mathcal {H}_\mathsf {M}\) is a nondisconnective handle and the connected matroid \(\mathsf {M}{\setminus } H_k=\mathsf {M}\vert _{F_{k1}}\) has a handle decomposition of length \(k1\). By induction, there are k (disjoint) nondisconnective handles \(H'_0,\dots ,H'_{k1}\in \mathcal {H}_{\mathsf {M}{\setminus } H_k}\). Since \(k\ge 3\) and handles are nonempty, \(H_k\cup H'_i\ne E\) for \(i=0,\dots ,k1\). For each \(i=0,\dots ,k1\), Lemma 2.9 now yields a nondisconnective handle \(H'_i\supseteq H''_i\in \mathcal {H}_\mathsf {M}\). Thus, \(\mathsf {M}\) has \(k+1\) (disjoint) nondisconnective handles \(H''_0,\dots ,H''_{k1},H_k\in \mathcal {H}_\mathsf {M}\). \(\square \)
We conclude this section with an observation.
Lemma 2.13
(Existence of circuits). Let \(\mathsf {M}\) be a connected matroid of rank \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\). Then there is a circuit \(C\in \mathcal {C}_\mathsf {M}\) of size \({\left C\right }\ge 3\).
Proof
Pick \(e\in E\). Since \(\mathsf {M}\) is connected, E is the union of all circuits \(e\in C\in \mathcal {C}_\mathsf {M}\). Suppose that there are only 2circuits. Then \(E={{\,\mathrm{cl}\,}}_\mathsf {M}(e)\) (see [26, Prop. 1.4.11.(ii)]) and hence \({{\,\mathrm{rk}\,}}\mathsf {M}=1\) (see (2.2)), a contradiction. \(\square \)
Configurations and realizations
Our objects of interest are not associated with a matroid itself but with a realization as defined in the following. All matroid operations we consider come with a counterpart for realizations. For graphic matroids, these agree with familiar operations on graphs (see §2.4).
Fix a field \(\mathbb {K}\) and denote the \(\mathbb {K}\)dualizing functor by
We consider a finite set E as a basis of the based \(\mathbb {K}\)vector space \(\mathbb {K}^E\) and denote by \(E^\vee =(e^\vee )_{e\in E}\) the dual basis of
By abuse of notation, we set \(S^\vee :=(e^\vee )_{e\in S}\) for any subset \(S\subseteq E\).
We consider configurations as defined by Bloch, Esnault and Kreimer (see [6, §1]).
Definition 2.14
(Configurations and realizations). Let E be a finite set. A \(\mathbb {K}\)vector subspace \(W\subseteq \mathbb {K}^E\) is called a configuration (over \(\mathbb {K}\)). It defines a matroid \(\mathsf {M}_W\) on E with independent sets
Let \(\mathsf {M}\) be a matroid and \(W\subseteq \mathbb {K}^E\) a configuration (over \(\mathbb {K}\)). If \(\mathsf {M}=\mathsf {M}_W\), then W is called a (linear) realization of \(\mathsf {M}\) and \(\mathsf {M}\) is called (linearly) realizable (over \(\mathbb {K}\)). A matroid is called binary if it is realizable over \(\mathbb {F}_2\). A configuration \(W\subseteq \mathbb {K}^E\) is called totally unimodular if \({{\,\mathrm{ch}\,}}\mathbb {K}=0\) and W admits a basis whose coefficient matrix with respect to E has all (maximal) minors in \({\left\{ 0,\pm 1\right\} }\). A matroid is called regular if it admits a totally unimodular realization. Equivalently, a regular matroid is realizable over every field (see [26, Thm. 6.6.3]).
Since \(E^\vee \vert _W\) generates \(W^\vee \), we have (see (2.14))
Remark 2.15
(Matroids and linear algebra). The notions in matroid theory (see §2.1) are derived from linear (in)dependence over \(\mathbb {K}\). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\). Pick a basis \(w=(w^1,\dots ,w^r)\) of W where \(r:={{\,\mathrm{rk}\,}}\mathsf {M}\) (see (2.15)). For each \(e\in E\), \(e^\vee \vert _W\) is then represented by the vector \((w^i_e)_i\in \mathbb {K}^r\) where \(w^i_e:=e^\vee (w^i)\) for \(i=1,\dots ,r\). Order \(E={\left\{ e_1,\dots ,e_n\right\} }\) and set \(w^i_j:=w^i_{e_j}\) for \(j=1,\dots ,n\). Then these vectors form the columns of the coefficient matrix \(A=(w^i_j)_{i,j}\in \mathbb {K}^{r\times n}\) of w. By construction, W is the row span of A. The matroid rank \({{\,\mathrm{rk}\,}}_\mathsf {M}(S)\) of any subset \(S\subseteq E\) now equals the \(\mathbb {K}\)linear rank of the submatrix of A with columns S (see (2.1) and (2.14)). An element \(e\in E\) is a loop in \(\mathsf {M}\) if and only if column e of A is zero; e is a coloop in \(\mathsf {M}\) if and only if column e is not in the span of the other columns.
Remark 2.16
(Classical configurations). Suppose that \(\mathsf {M}_W\) has no loops, that is, \(e^\vee \vert _W\ne 0\) for each \(e\in E\). Then the images of the \(e^\vee \vert _W\) in \(\mathbb {P}W^\vee \) form a projective point configuration in the classical sense (see [19]). Dually, the hyperplanes \(\ker (e^\vee )\cap W\) form a hyperplane arrangement in W (see [25]), which is an equivalent notion in this case.
We fix some notation for realizations of basic matroid operations. Any subset \(S\subseteq E\) gives rise to an inclusion and a projection
of based \(\mathbb {K}\)vector spaces.
Definition 2.17
(Realizations of matroid operations). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\), and let \(F\subseteq E\) be any subset.

(a)
The restriction configuration (see (2.16))
$$\begin{aligned} W\vert _F&:=\pi _F(W)\subseteq \mathbb {K}^F\\&\cong (W+\mathbb {K}^{E{\setminus } F})/\mathbb {K}^{E{\setminus } F}\cong W/(W\cap \mathbb {K}^{E{\setminus } F}) \end{aligned}$$realizes the restriction matroid \(\mathsf {M}\vert _F\).

(b)
The deletion configuration
$$\begin{aligned} W{\setminus } F:=W\vert _{E{\setminus } F} \end{aligned}$$realizes the deletion matroid \(\mathsf {M}{\setminus } F\). We write \(W{\setminus } e:=W\setminus {\left\{ e\right\} }\) for \(e\in E\).

(c)
The contraction configuration
$$\begin{aligned} W/F:=W\cap \mathbb {K}^{E{\setminus } F}\subseteq \mathbb {K}^{E{\setminus } F} \end{aligned}$$realizes the contraction matroid \(\mathsf {M}/F\).

(d)
The dual configuration (see (2.13))
$$\begin{aligned} W^\perp :=(\mathbb {K}^E/W)^\vee \subseteq \mathbb {K}^{E^\vee } \end{aligned}$$realizes the dual matroid \(\mathsf {M}^\perp \).

(e)
Any \(0\ne \varphi \in W^\vee \) defines an elementary quotient configuration
$$\begin{aligned} W_\varphi :=\ker \varphi \subseteq \mathbb {K}^E. \end{aligned}$$
Remark 2.18
Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\).

(a)
An element \(e\in E\) is a loop or coloop in \(\mathsf {M}\) if and only if \(W\subseteq \mathbb {K}^{E{\setminus }{\left\{ e\right\} }}\) or \(W=(W{\setminus } e)\oplus \mathbb {K}^{\left\{ e\right\} }\), respectively. In both cases, \(W{\setminus } e=W/e\subseteq \mathbb {K}^{E{\setminus }{\left\{ e\right\} }}\).

(b)
For \(0\ne \varphi \in W^\vee \), pick \(w\in W{\setminus } W_\varphi \) and \(e\notin E\). Consider the configuration
$$\begin{aligned} W_{\varphi ,w}:=W_\varphi \oplus \mathbb {K}\cdot (w+e)\subseteq \mathbb {K}^{E\sqcup {\left\{ e\right\} }}. \end{aligned}$$Then \(W_{\varphi ,w}{\setminus } e=W\) and \(W_{\varphi ,w}/e=W_\varphi \). By definition, \(\mathsf {M}_{W_\varphi }\) is therefore an elementary quotient of \(\mathsf {M}_W\); it can be characterized in terms of the notion of a modular cut (see [21, §5.5] and [26, §7.3]). \(\square \)
Lemma 2.19
(Lift of direct sums to realizations). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\). Suppose that \(\mathsf {M}=\mathsf {M}_1\oplus \mathsf {M}_2\) decomposes with underlying partition \(E=E_1\sqcup E_2\). Then \(W=W_1\oplus W_2\) where \(W_i:=\mathsf {M}/E_j\subseteq \mathbb {K}^{E_i}\) realizes \(\mathsf {M}_i=\mathsf {M}\vert _{E_i}\) for \({\left\{ i,j\right\} }={\left\{ 1,2\right\} }\).
Proof
By definition (see Definition 2.17.(a) and (c)),
By the direct sum hypothesis, \(W_i\) and \(W\vert _{E_i}\) realize the same matroid (see (2.3), (2.4) and (2.6))
Thus, \(\dim W_i=\dim (W\vert _{E_i})\) for \(i=1,2\) (see (2.15)) and the claim follows. \(\square \)
Example 2.20
(Realizations of uniform matroids). Let \(W\subseteq \mathbb {K}^E\) be the row span of a matrix \(A\in \mathbb {K}^{r\times n}\) (see Remark 2.15). If A is generic in the sense that all maximal minors of A are nonzero, then W realizes the uniform matroid \(\mathsf {U}_{r,n}\) (see Example 2.2).
Graphic matroids
Configurations arising from graphs are the most prominent examples for our results. In this subsection, we review this construction and discuss important examples such as prism, wheel and whirl matroids.
A graph \(G=(V,E)\) is a pair of finite sets V of vertices and E of (unoriented) edges where each edge \(e\in E\) is associated with a set of one or two vertices in V. This allows for multiple edges between pairs of vertices, and loops at vertices.
A graph G determines a graphic matroid \(\mathsf {M}_G\) on the edge set E. Its independent sets are the forests and its circuits the simple cycles in G. Any graphic matroid comes from a (nonunique) connected graph (see [26, Prop. 1.2.9]). Unless specified otherwise, we therefore assume that G is connected. Then the bases of \(\mathsf {M}_G\) are the spanning trees of G (see [26, p. 18]),
Remark 2.21
(Graph and matroid connectivity). A vertex cut of a graph \(G=(V,E)\) is a subset of V whose removal (together with all incident edges) disconnects G. If G has at least one pair of distinct nonadjacent vertices, then G is called kconnected if k is the minimal size of a vertex cut. Otherwise, G is \(({\left V\right }1)\)connected by definition. Suppose that \({\left V\right }\ge 3\). Then \(\mathsf {M}_G\) is (2)connected if and only if G is 2connected and loopless (see [26, Prop. 4.1.7]). Provided that \({\left E\right }\ge 4\), \(\mathsf {M}_G\) is 3connected if and only if G is 3connected and simple (see [26, Prop. 8.1.9]).
Example 2.22
(Prism matroid as graphic matroid). The prism matroid (see Definition 2.1) is associated with the (2, 2, 2)theta graph in Fig. 2. In particular it is 3connected as witnessed by the minimal vertex cut \({\left\{ v_1,v_2,v_3\right\} }\) (see Remark 2.21).
Graphic matroids have realizations derived from the edgevertex incidence matrix of the graph (see [6, §2]). A choice of orientation on the edge set E turns the graph G into a CWcomplex. This gives rise to an exact sequence
where \(H_\bullet :=H_\bullet (G,\mathbb {K})\) denotes the graph homology of G over \(\mathbb {K}\). The dual exact sequence
involves the graph cohomology \(H^\bullet :=H^\bullet (G,\mathbb {K})\) of G over \(\mathbb {K}\).
Definition 2.23
(Graph configurations). We call the image
of \(\delta ^\vee \) the graph configuration of the graph G over \(\mathbb {K}\). Note that it is independent of the orientation chosen to define \(\delta \) in (2.18).
For any \(S\subseteq E\), the sequence (2.18) induces a short exact sequence
By definition of \(\mathsf {M}_G\) and \(\mathsf {M}_{W_G}\) (see Definition 2.14) and since \(H_1\) is generated by indicator vectors of (simple) cycles, we have
which implies that
The configuration \(W_G\) is totally unimodular if \({{\,\mathrm{ch}\,}}\mathbb {K}=0\) (see [26, Lem. 5.1.4]) which makes \(\mathsf {M}_G\) a regular matroid. By construction, \(W_G^\perp =H_1\subseteq \mathbb {K}^E\) realizes the dual matroid \(\mathsf {M}_G^\perp \) (see Definition 2.17.(d)).
Example 2.24
(Configuration of the (2, 2, 2)theta graph). With the orientation of the (2, 2, 2)theta graph G depicted in Fig. 2, the map \(\delta ^\vee \) in (2.19) is represented by the transpose of the matrix
Its rows generate the graph configuration \(W_G\) realizing the prism matroid (see Example 2.22).
Lemma 2.25
(Characterization of the prism matroid). Let \(\mathsf {M}\) be a connected matroid on \(E={\left\{ e_1,\dots ,e_6\right\} }\) with \({\left E\right }=6\) whose handle partition
is made of 3 maximal 2handles (see Example 2.7 and Lemma 2.10). Then \(\mathsf {M}\) is the prism matroid (see Definition 2.1). Up to scaling E, it has the unique realization \(W\subseteq \mathbb {K}^E\) with basis
the graph configuration of the (2, 2, 2)theta graph (see Example 2.24).
Proof
Each circuit \(C\in \mathcal {C}_\mathsf {M}\) is a (nonempty) disjoint union of \(H_1,H_2,H_3\) (see Definition 2.3). By Lemma 2.4.(b), no \(H_i\) is a circuit, but each \(H_i\) is properly contained in one. By hypothesis, E is not a maximal handle and hence \(E\not \in \mathcal {C}_\mathsf {M}\). Up to renumbering \(H_1,H_2,H_3\), this yields circuits \(H_2\sqcup H_3\) and \(H_1\sqcup H_3\). By the strong circuit elimination axiom (see [26, Prop. 1.4.12]), there is a third circuit \(H_1\sqcup H_2\). Then
identifies with the circuits of the prism matroid. It follows that \(\mathsf {M}\) must be the prism matroid.
Let \(W\subseteq \mathbb {K}^E\) be any realization of \(\mathsf {M}\). Then \(\dim W={{\,\mathrm{rk}\,}}\mathsf {M}=4\) (see (2.15) and (2.17)). Pick a basis \(w=(w^1,\dots ,w^4)\) of W and denote by \(A=(w^i_j)_{i,j}\) the coefficient matrix (see Remark 2.15). We may assume that columns 2, 4, 6, 5 of A form an identity matrix. Since \(C_1\) and \(C_2\) are circuits, \(w^1_3=0\ne w^2_3\) and \(w^2_1=0\ne w^1_1\). Thus,
Since \(C_3\) is a circuit, suitably replacing \(w^3,w^4\in {\left\langle w^3,w^4\right\rangle }\), reordering \(H_3\) and scaling \(e_1,e_3\) makes
where \(w^1_1,w^2_3,w^3_5\ne 0\). Now suitably scaling first \(w^1,w^2,w^3\) and then \(e_2,e_4,e_6\) makes
Now \(w=(w^1,\dots ,w^4)\) is the desired basis. \(\square \)
The following classes of matroids play a distinguished role in connection with 3connectedness.
Example 2.26
(Wheels and whirls). For \(n\ge 2\), the wheel graph \(W_n\) in Fig. 3 is obtained from an ncycle, the “rim,” by adding an additional vertex and edges, the “spokes,” joining it to each vertex in the rim. There is a partition of the set of edges
into the set S of spokes and the set R of edges in the rim. The symmetry suggests to use a cyclic index set \(\mathbb {Z}_n:=\mathbb {Z}/n\mathbb {Z}={\left\{ 1,\dots ,n\right\} }\).
For \(n\ge 3\), the wheel matroid is the graphic matroid \(\mathsf {W}_n:=\mathsf {M}_{W_n}\) on E. For \(n\ge 2\), the whirl matroid is the (nongraphic) matroid on E obtained from \(\mathsf {M}_{W_n}\) by relaxation of the rim R, that is,
In terms of circuits, this means that
The matroids \(\mathsf {W}_n\) and \(\mathsf {W}^n\) are 3connected (see [26, Exa. 8.4.3]) of rank
For each \(i\in \mathbb {Z}_n\), \({\left\{ s_i,r_i,s_{i+1}\right\} }\) is a triangle and \({\left\{ r_i,r_{i+1},s_{i+1}\right\} }\) a triad. Conversely, this property enforces \(\mathsf {M}\in {\left\{ \mathsf {W}_n,\mathsf {W}^n\right\} }\) for any connected matroid \(\mathsf {M}\) on E (see [29, (6.1)]).
We describe all realizations of wheels and whirls up to equivalence. In particular, we recover the wellknown fact that whirls are not binary.
Lemma 2.27
(Realizations of wheels and whirls). Let \(W\subseteq \mathbb {K}^E\) be any realization of \(\mathsf {M}\in {\left\{ \mathsf {W}_n,\mathsf {W}^n\right\} }\). Up to scaling \(E=S\sqcup R\), W has a basis
where \(t=1\) if \(\mathsf {M}=\mathsf {W}_n\), and \(t\in \mathbb {K}{\setminus }{\left\{ 0,1\right\} }\) if \(\mathsf {M}=\mathsf {W}^n\).
Proof
Since \(S\in \mathcal {B}_\mathsf {M}\), we may assume that the coefficients of \(s_j\) in \(w^i\) form an identity matrix, that is, \(w^i_{s_j}=\delta _{i,j}\). The triangle \({\left\{ s_j,r_j,s_{j+1}\right\} }\) then forces \(w^j_{r_j},w^{j+1}_{r_j}\ne 0\) and \(w^i_{r_j}=0\) for all \(i\in \mathbb {Z}_n{\setminus }{\left\{ j,j+1\right\} }\). Suitably scaling \(r_1,w^2,r_2,w^3,\dots ,r_{n1},w^n,s_1,\dots ,s_n\) successively yields (2.20). The claim on t follows from \(R\in \mathcal {C}_{\mathsf {W}_n}\) and \(R\in \mathcal {B}_{W^n}\), respectively. \(\square \)
Configuration polynomials and forms
In this section, we develop Bloch’s strategy of putting graph polynomials into the context of configuration polynomials and configuration forms. We lay the foundation for an inductive proof of our main result using a handle decomposition. In the process, we generalize some known results on graph polynomials to configuration polynomials.
Configuration polynomials
To prepare the definition of configuration polynomials we introduce some notation.
Let \(W\subseteq \mathbb {K}^E\) be a configuration, and let \(S\subseteq E\) be any subset. Compose the associated inclusion map with \(\pi _S\) to a map (see (2.16))
Fix an isomorphism
and set \(c_0:={{\,\mathrm{id}\,}}_\mathbb {K}\) for the zero vector space. Any basis of W gives rise to such an isomorphism and any two such isomorphisms differ by a nonzero multiple \(c\in \mathbb {K}^*\). Up to sign or ordering E, we identify
as based vector spaces. Suppose that \({\left S\right }=\dim W\). Then the determinant
is defined up to sign. Its square
is defined up to a factor \(c^2\) for some \(c\in \mathbb {K}^*\) independent of S. Note that \(\det \alpha _{0,\emptyset }={{\,\mathrm{id}\,}}_\mathbb {K}\) and hence \(c_{0,\emptyset }=1\). By definition (see (2.14)),
Remark 3.1
(Compatibility of coefficients with restriction). Let \(W\subseteq \mathbb {K}^E\) be a configuration, and let \(S\subseteq F\subseteq E\) with \({\left S\right }=\dim W\). Then the maps (3.1) for W and \(W\vert _F\) form a commutative diagram
and hence \(c_{W,S}=c^2\cdot c_{W\vert _F,S}\) for some \(c\in \mathbb {K}^*\) independent of S.
Consider the dual basis \(E^\vee =(e^\vee )_{e\in E}\) of E as coordinates on \(\mathbb {K}^E\),
Given an enumeration of \(E={\left\{ e_1,\dots ,e_n\right\} }\), we write
For any subset \(S\subseteq E\), we set
Definition 3.2
(Configuration polynomials). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\). Then the configuration polynomial of W is (see (3.5))
Remark 3.3
(Welldefinedness of configuration polynomials). Any two isomorphisms \(c_W\) (see (3.2)) differ by a nonzero multiple \(c\in \mathbb {K}^*\). Using the isomorphism \(c\cdot c_W\) in place of \(c_W\) replaces \(\psi _W\) by \(c^2\cdot \psi _W\). In other words, \(\psi _W\) is welldefined up to a nonzero constant square factor. Whenever \(\psi _W\) occurs in a formula, we mean that the formula holds true for a suitable choice of such a factor.
Remark 3.4
(Equivalence of configuration polynomials). Dividing \(e\in E\) by \(c\in \mathbb {K}^*\) multiplies both \(x_e=e^\vee \) (see Remark 2.16) and the identifications (3.3) with \(e\in S\) by c. For each \(e\in B\in \mathcal {B}_\mathsf {M}\), this multiplies \(c_{W,B}\) by \(c^2\) and \(x^B\) by c. This is equivalent to substituting \(c^3\cdot x_e\) for \(x_e\) in \(\psi _W\). Scaling E thus results in scaling x in \(\psi _W\).
However, dropping the equality (3.7) and scaling \(e\in E\) for fixed \(x_e\) replaces W in \(\psi _W\) by a projectively equivalent realization (see [26, §6.3]). If \(\mathsf {M}\) is binary, then all realizations of \(\mathsf {M}\) over \(\mathbb {K}\) are projectively equivalent (see [26, Prop. 6.6.5]). The corresponding configuration polynomials are geometrically equivalent in this case. In general, however, there are geometrically different configuration polynomials for fixed \(\mathsf {M}\) and \(\mathbb {K}\) (see Example 5.3).
Remark 3.5
(Degree of configuration polynomials). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\). Then (see (2.15) and (3.6))
In particular, \(\psi _W\ne 0\), and \(\psi _W=1\) if and only if \({{\,\mathrm{rk}\,}}\mathsf {M}=0\). By definition, \(\psi _W\) is independent of (divided by) \(x_e\) if and only if \(e\in E\) is a (co)loop in \(\mathsf {M}\).
Remark 3.6
(Matroid polynomials and regularity). For any matroid \(\mathsf {M}\), not necessarily realizable, there is a matroid (basis) polynomial
If \(\mathsf {M}\) is regular, then \(\psi _W=\psi _\mathsf {M}\) for any totally unimodular realization W of \(\mathsf {M}\) over \(\mathbb {K}\). Conversely, this equality for some realization W over \(\mathbb {K}\) with \({{\,\mathrm{ch}\,}}\mathbb {K}=0\) establishes regularity of \(\mathsf {M}\). For regular \(\mathsf {M}\), all configuration polynomials over \(\mathbb {K}\) are geometrically equivalent (see Remark 3.4). In general, however, \(\psi _W\) and \(\psi _\mathsf {M}\) are geometrically different (see Example 5.2).
Example 3.7
(Configuration polynomials of uniform matroids). Let \(W\subseteq \mathbb {K}^E\) be a realization of a uniform matroid \(\mathsf {M}=\mathsf {U}_{r,n}\) (see Example 2.20).

(a)
Suppose that \(\mathsf {M}=\mathsf {U}_{n,n}\) is a free matroid. Then \(E\in \mathcal {B}_\mathsf {M}\) and
$$\begin{aligned} \psi _W=x^E \end{aligned}$$is the elementary symmetric polynomial of degree n in n variables.

(b)
Suppose that \(\mathsf {M}=\mathsf {U}_{n1,n}\) is a circuit. Then \(E\in \mathcal {C}_\mathsf {M}\) and by Remark 3.1 and (a)
$$\begin{aligned} \psi _{W}=\sum _{e\in E}\psi _{W{\setminus } e},\quad \psi _{W{\setminus } e}=x^{E\setminus {\left\{ e\right\} }}. \end{aligned}$$A priori, substituting \(x^{E{\setminus }{\left\{ e\right\} }}\) for \(\psi _{W{\setminus } e}\) in \(\psi _{W}\) is invalid (see Remark 3.3). However, this can be achieved as follows: Ordering \(E={\left\{ e_1,\dots ,e_n\right\} }\), W has a basis \(w^i=e_i+c_i\cdot e_n\) with \(c_i\in \mathbb {K}^*\) where \(i=1,\dots ,n1\). Scaling first \(w^1,\dots ,w^{n1}\) and then \(e_1,\dots ,e_{n1}\) makes \(c_1=\dots =c_{n1}=1\). This turns \(\psi _W\) into
$$\begin{aligned} \psi _W=\sum _{e\in E}x^{E{\setminus }{\left\{ e\right\} }}, \end{aligned}$$the elementary symmetric polynomial of degree \(n1\) in n variables.

(c)
If \(\mathsf {M}=\mathsf {U}_{n2,n}\), then \(\mathsf {M}\) has \(n\atopwithdelims ()n2\) bases, and \(\psi _W\) has \(n\atopwithdelims ()n2\) monomials whose coefficients depend on the choice of W. For instance, the row span W of the matrix
$$\begin{aligned} \begin{pmatrix} 1 &{} \quad 0 &{} \quad 1 &{} \quad 1\\ 0 &{} \quad 1 &{} \quad 1 &{} \quad 1 \end{pmatrix} \end{aligned}$$realizes \(\mathsf {U}_{2,4}\) and
$$\begin{aligned} \psi _W=x_1x_2+x_1x_3+x_1x_4+x_2x_3+x_2x_4+4x_3x_4. \end{aligned}$$Realizations of \(\mathsf {U}_{2,n}\) are treated in Example 5.4. \(\square \)
In the following, we put matroid connectivity in correspondence with irreducibility of configuration polynomials.
Proposition 3.8
(Connectedness and irreducibility). Let \(\mathsf {M}\) be a matroid of rank \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 1\) with realization \(W\subseteq \mathbb {K}^E\). Then \(\mathsf {M}\) is connected if and only if \(\mathsf {M}\) has no loops and \(\psi _W\) is irreducible. In particular, if \(\mathsf {M}=\bigoplus _{i=1}^n\mathsf {M}_i\) with connected components \(\mathsf {M}_i\) and induced decomposition \(W=\bigoplus _{i=1}^nW_i\) (see Lemma 2.19), then \(\psi _W=\prod _{i=1}^n\psi _{W_i}\) where \(\psi _{W_i}\) is irreducible if \({{\,\mathrm{rk}\,}}\mathsf {M}_i\ge 1\), and \(\psi _{W_i}=1\) otherwise.
Proof
First suppose that \(\mathsf {M}=\mathsf {M}_1\oplus \mathsf {M}_2\) is disconnected with underlying proper partition \(E=E_1\sqcup E_2\). By Lemma 2.19, \(W=W_1\oplus W_2\) where \(W_i\subseteq \mathbb {K}^{E_i}\) realizes \(\mathsf {M}_i\). Then \(\alpha _{W,B}=\alpha _{W_1,B_1}\oplus \alpha _{W_2,B_2}\) and hence \(c_{W,B}=c_{W_1,B_1}\cdot c_{W_2,B_2}\) for all \(B=B_1\sqcup B_2\in \mathcal {B}_\mathsf {M}\) where \(B_i\in \mathcal {B}_{\mathsf {M}_i}\) for \(i=1,2\) (see (2.3)). It follows that \(\psi _W=\psi _{W_1}\cdot \psi _{W_2}\). This factorization is proper if \(\mathsf {M}\) and hence each \(\mathsf {M}_i\) has no loops (see Remark 3.5). Thus, \(\psi _W\) is reducible in this case.
Suppose now that \(\psi _W\) is reducible. Then
with \(\psi _i\) homogeneous nonconstant for \(i=1,2\). Since \(\psi _W\) is a linear combination of squarefree monomials (see Definition 3.2), this yields a proper partition \(E=E_1\sqcup E_2\) such that \(\psi _i\in \mathbb {K}[x_{E_i}]\) for \(i=1,2\). Set
Each basis \(B\in \mathcal {B}_\mathsf {M}\) indexes a monomial \(x^B\) in \(\psi _W\) (see (3.6)). Set \(B_i:=B\cap E_i\in \mathcal {I}_{\mathsf {M}_i}\) for \(i=1,2\) (see (2.4)). Then \(x^B=x^{B_1}\cdot x^{B_2}\) where \(x^{B_i}\) is a monomial in \(\psi _i\) for \(i=1,2\). By homogeneity of \(\psi _i\), \(B_i\in \mathcal {B}_{\mathsf {M}_i}\) for \(i=1,2\) and hence \(B=B_1\sqcup B_2\in \mathcal {B}_{\mathsf {M}_1\oplus \mathsf {M}_2}\) (see (2.3)). It follows that \(\mathcal {B}_\mathsf {M}\subseteq \mathcal {B}_{\mathsf {M}_1\oplus \mathsf {M}_2}\).
Conversely, let \(B=B_1\sqcup B_2\in \mathcal {B}_{\mathsf {M}_1\oplus \mathsf {M}_2}\) where \(B_i\in \mathcal {B}_{\mathsf {M}_i}\) for \(i=1,2\). Then \(B_i=B_i'\cap E_i\) for some \(B_i'\in \mathcal {B}_\mathsf {M}\) for \(i=1,2\) (see (2.4) and (3.9)). As above, \(x^{B_i}\) is a monomial in \(\psi _i\) for \(i=1,2\). Then \(x^B=x^{B_1}\cdot x^{B_2}\) is a monomial in \(\psi _W\) and hence \(B\in \mathcal {B}_\mathsf {M}\) (see (3.6)). It follows that \(\mathcal {B}_\mathsf {M}\supseteq \mathcal {B}_{\mathsf {M}_1\oplus \mathsf {M}_2}\) as well.
So \(\mathsf {M}=\mathsf {M}_1\oplus \mathsf {M}_2\) is a proper decomposition and \(\mathsf {M}\) is disconnected.
This proves the equivalence and the particular claims follow. \(\square \)
We use the following wellknown fact from linear algebra.
Remark 3.9
(Determinant formula). Consider a short exact sequence of finite dimensional \(\mathbb {K}\)vector spaces
Abbreviate \(\bigwedge V:=\bigwedge ^{\dim V}V\). There is a unique isomorphism
that fits into a commutative diagram of canonical maps
Tensored with
respectively, it induces identifications
Consider a commutative diagram of finite dimensional \(\mathbb {K}\)vector spaces with short exact rows
Then the above identifications for both rows fit into a commutative diagram
The following result of Bloch, Esnault and Kreimer describes the behavior of configuration polynomials under duality (see [6, Prop. 1.6]).
Proposition 3.10
(Dual configuration polynomials). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\). For a suitable choice of \(c_W\) (see (3.2)),
for all \(S\subseteq E\) of size \({\left S\right }={{\,\mathrm{rk}\,}}\mathsf {M}\). In particular,
Proof
Let \(S\subseteq E\) be of size \({\left S\right }={{\,\mathrm{rk}\,}}\mathsf {M}\). Then \(S\in \mathcal {B}_\mathsf {M}\) if and only if \(S^\perp \in \mathcal {B}_{\mathsf {M}^\perp }\) (see Remark 3.3). We may assume that this is the case as otherwise both determinants are zero. Then there is a commutative diagram with exact rows
where the middle isomorphism is induced by (2.8). This yields a commutative diagram (Remark 3.9 and (2.15))
Using (3.3), we may drop \(\bigwedge ^{\left E\right }\mathbb {K}^E\) and \(\bigwedge ^{\left E\right }\mathbb {K}^{E^\vee }\). A suitable choice of \(c_W\) turns the upper isomorphism into an equality. The claim follows by definition (see (3.4) and Definition 3.2). \(\square \)
The coefficients of the configuration polynomial satisfy the following restriction–contraction formula.
Lemma 3.11
(Restriction–contraction for coefficients). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\), and let \(F\subseteq E\) be any subset. For any basis \(B\in \mathcal {B}_\mathsf {M}\), \(B\cap F\in \mathcal {B}_{\mathsf {M}\vert _F}\) if and only if \(B{\setminus } F\in \mathcal {B}_{\mathsf {M}/F}\). In this case,
where \(c\in \mathbb {K}^*\) is independent of B.
Proof
The equivalence for \(B\in \mathcal {B}_\mathsf {M}\) holds by definition of matroid contraction (see (2.6)). For any such B, there is a commutative diagram with exact rows (see Definition 2.17.(a) and (c))
Taking exterior powers yields (see Remark 3.9 and (2.15))
\(\square \)
The following result describes the behavior of configuration polynomials under deletion–contraction. It is the basis for our inductive approach to Jacobian schemes of configuration polynomials. The statement on \(\partial _e\psi _W\) was proven by Patterson (see [27, Lem. 4.4]).
Proposition 3.12
(Deletion–contraction for configuration polynomials). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\), and let \(e\in E\). Then
where \(\psi _{W\vert _e}=x_e\) if e is not a loop in \(\mathsf {M}\). In particular,
Proof
Decompose
The second sum in (3.10) is nonzero if and only if e is not a loop. Suppose that this is the case. Then \(\mathsf {M}\vert _e\) is free with basis \({\left\{ e\right\} }\) and \(\psi _{W\vert _e}=x_e\) by Remark 3.7.(a). By Lemma 3.11 applied to \(F={\left\{ e\right\} }\), the second sum in (3.10) then equals (see (2.6) and Remark 3.3)
for some \(c\in \mathbb {K}^*\). The first sum in (3.10) is nonzero if and only if e is not a coloop. By Lemma 3.11 applied to \(F=E{\setminus }{\left\{ e\right\} }\), it equals in this case (see (2.4) and Remark 3.3)
for some \(c\in \mathbb {K}^*\). If e is a (co)loop, then \(W/e=W{\setminus } e\) (see Remark 2.18.(a)). The claimed formulas follow. \(\square \)
The following formula relates configuration polynomials with deletion and contraction of handles. It is the starting point for our description of generic points of Jacobian schemes of configuration hypersurfaces in terms of handles.
Corollary 3.13
(Configuration polynomials and handles). Let \(W\subseteq \mathbb {K}^E\) be a realization of a connected matroid \(\mathsf {M}\) on E, and let \(E\ne H\in \mathcal {H}_\mathsf {M}\) be a proper handle. Then
In particular, after suitably scaling H,
Proof
By Lemma 2.4.(b), \(H\in \mathcal {C}_{\mathsf {M}/(E{\setminus } H)}\) and hence (3.12) by Example 3.7.(b). By Lemma 2.4.(b) (see (2.4)), \(\mathsf {M}\vert _H\) is free, and equalities (3.13) follows from Example 3.7.(a). Equality (3.14) follows from (3.11), (3.12) and Example 3.7.(b). It remains to prove equality (3.11).
We proceed by induction on \({\left H\right }\). Let \(h\in H\) and set \(H':=H{\setminus }{\left\{ h\right\} }\). Since \(\mathsf {M}\) is connected, it has no (co)loops and hence
by Proposition 3.12. If \({\left H\right }=1\), then \(H\in \mathcal {C}_{\mathsf {M}/(E{\setminus } H)}\) implies that \({{\,\mathrm{rk}\,}}(\mathsf {M}/(E{\setminus } h))=0\) and hence \(\psi _{W/(E{\setminus } h)}=1\) (see Remark 3.5). Suppose now that \({\left H\right }\ge 2\). By Lemma 2.4.(b) and (c), \(\mathsf {M}\vert _{H'}\) is free and \(H'\) consists of coloops in \(\mathsf {M}{\setminus } h\). Iterating Proposition 3.12 thus yields
By Lemma 2.4.(d), the set \(H'\) is a proper handle in the connected matroid \(\mathsf {M}/h\). By Lemma 2.4.(c), h is a coloop in \(\mathsf {M}{\setminus } H'\) and hence
by Remark 2.18.(a). By the induction hypothesis,
By Lemma 2.4.(b), \(\mathsf {M}\vert _H\) and \(\mathsf {M}\vert _{H{\setminus }{\left\{ h'\right\} }}\) are free. Iterating Proposition 3.12 thus yields
Using equalities (3.12) and (3.18), equality (3.11) is obtained by substituting (3.16) and (3.17) into (3.15) (see Remark 3.3). \(\square \)
The following result describes the behavior of configuration polynomials when passing to an elementary quotient.
Proposition 3.14
(Configuration polynomials of quotients). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\), and let \(0\ne \varphi \in W^\vee \). Then
where \(\tilde{\varphi }=(\tilde{\varphi }_e)_{e\in E}\in (\mathbb {K}^E)^\vee \) is any lift of \(\varphi \) with a sign ± determined by a Laplace expansion.
Proof
Set \(V:=W^\perp \) and \(V_\varphi :=W_\varphi ^\perp \) and consider the commutative diagram with short exact rows and columns
Dualizing and identifying the two copies of \(\mathbb {K}\) by the Snake Lemma yields a commutative diagram with short exact rows and columns
By Remark 3.9 and with a suitable choice of \(c_V\) (see Remark 3.3), the right vertical short exact sequence in (3.19) gives rise to a commutative square
Let \(S'\subseteq E^\vee \) with \({\left S'\right }=\dim V_\varphi ={{\,\mathrm{rk}\,}}\mathsf {M}^\perp +1\) and denote (see (2.8))
Due to (3.19) the maps \(\alpha _{V_\varphi ,S'}\) (see (3.1)) and
agree after applying \(\bigwedge ^{{{\,\mathrm{rk}\,}}\mathsf {M}^\perp +1}\). Laplace expansion thus yields
Let \(S\subseteq E\) with \({\left S\right }=\dim W_\varphi ={{\,\mathrm{rk}\,}}\mathsf {M}1\) and \(S'=S^\perp \). Then Proposition 3.10 yields
\(\square \)
Graph polynomials
We continue the discussion of graphic matroids from §2.4 and consider their configuration polynomials.
Definition 3.15
(Graph polynomials). The (first) Kirchhoff polynomial of a graph G over \(\mathbb {K}\) is the polynomial
Replacing \(x^T\) by \(x^{E{\setminus } T}\) defines the (first) Symanzik polynomial \(\psi _G^\perp \) of a graph G over \(\mathbb {K}\). We refer to \(\psi _G\) and \(\psi _G^\perp \) as (first) graph polynomials.
By (2.17), we have \(\psi _G=\psi _W\) for any totally unimodular realization W of \(\mathsf {M}_G\). In particular, this yields the following result of Bloch, Esnault and Kreimer (see [6, Prop. 2.2] and Proposition 3.10).
Proposition 3.16
(Graph polynomials as configuration polynomials). The graph polynomials
are the configuration polynomials of the graph configuration and of its dual (see Definition 2.23). \(\square \)
Example 3.17
(Graph polynomial of the prism). For the unique realization \(W=W_G\) of the prism matroid (see Lemma 2.25),
is the Kirchhoff polynomial of the (2, 2, 2)theta graph G (see Fig. 2).
Let \(G=(E,V)\) be a graph. A 2forest in G is an acyclic subgraph T of G with \({\left V\right }2\) edges. Any such \(T={\left\{ T_1,T_2\right\} }\) has 2 connected components \(T_1\) and \(T_2\). We denote by \(\mathcal {T}^2_G\) the set of all 2forests in G.
Definition 3.18
(Second graph polynomials). The second Kirchhoff polynomial of a graph G over \(\mathbb {K}\) is the polynomial
depending on a momentum \(0\ne p\in \ker \sigma \) for G over \(\mathbb {K}\) (see (2.18)). Note that
and hence, the coefficient \(m_{T_1}(p)^2\in \mathbb {K}\) of \(\psi _G(p)\) is welldefined.
Replacing the 2forests \(T_1\sqcup T_2\) by cut sets \(E{\setminus }(T_1\sqcup T_2)\) defines the second Symanzik polynomial \(\psi _G^\perp (p)\) of a graph G over \(\mathbb {K}\) (see [27, Def. 3.6]). We refer to \(\psi _G(p)\) and \(\psi _G^\perp (p)\) as second graph polynomials.
The following reformulation of a result of Patterson realizes second graph polynomials as configuration polynomials of a (dual) elementary quotient (see [27, Prop. 3.3] and Proposition 3.10). Patterson’s proof makes the general formula in Proposition 3.14 explicit in case of graph configurations (see [27, Lem. 3.4]).
Proposition 3.19
(Second graph polynomials as configuration polynomials). The second graph polynomials
are the configuration polynomials of the quotient of the graph configuration by a momentum and of its dual (see Definitions 2.17.(d) and (e) and 2.23). \(\square \)
Configuration forms
The configuration form yields an equivalent definition of the configuration polynomial as a determinant of a symmetric matrix with linear entries. Its second degeneracy locus turns out to be the nonsmooth locus of the hypersurface defined by the corresponding configuration polynomial.
Definition 3.20
(Configuration forms). Let \(\mu _\mathbb {K}\) denote the multiplication map of \(\mathbb {K}\). Consider the generic diagonal bilinear form on \(\mathbb {K}^E\),
Let \(W\subseteq \mathbb {K}^E\) be a configuration. Then the configuration (bilinear) form of W is the restriction of \(Q_{\mathbb {K}^E}\) to W,
Alternatively, it can be seen as the composition of canonical maps
where \([x]\) means \(\otimes \mathbb {K}[x]\). For \(k=0,\dots ,r:=\dim W\), it defines a map
Its image is the kth Fitting ideal \({{\,\mathrm{Fitt}\,}}_k{{\,\mathrm{coker}\,}}Q_W\) (see [16, §20.2]) and defines the \(k1\)st degeneracy scheme of \(Q_W\). We set
Note the different fonts used for \(M_W\) and \(\mathsf {M}_W\) (see Definition 2.14).
Remark 3.21
(Configuration forms as matrices). With respect to a basis \(w=(w^1,\dots ,w^r)\) of W, \(Q_W\) becomes a matrix of Hadamard products (see Remark 2.15)
Let \(Q^{i,j}\) denote the submaximal minor of a square matrix Q obtained by deleting row i and column j. Then
Any basis of W can be written as \(w'=Uw\) for some \(U\in {{\,\mathrm{Aut}\,}}_\mathbb {K}W\). Then
and the \(Q_{w'}^{i,j}\) become \(\mathbb {K}\)linear combinations of the \(Q_w^{i,j}\). We often consider \(Q_W\) as a matrix \(Q_w\) determined up to conjugation.
Remark 3.22
(Configuration forms and basis scaling). Scaling E results in scaling x in \(Q_W\) and in \(M_W\) (see Remark 3.4).
Bloch, Esnault and Kreimer defined \(\psi _W\) in terms of \(Q_W\) (see [6, Lem. 1.3]).
Lemma 3.23
(Configuration polynomial from configuration form). For any configuration \(W\subseteq \mathbb {K}^E\), the configuration polynomial
is the determinant of the configuration form (see Remarks 3.3 and 3.21). \(\square \)
Example 3.24
(Configuration form of the prism realization). Consider the realization W of the prism matroid with basis given in Lemma 2.25. Then the corresponding matrix of \(Q_W\) reads (see Remark 3.21)
Lemma 3.23 recovers the polynomial \(\det Q_W=\psi _W\) in Example 3.17.
The following result describes the behavior of Fitting ideals of configuration forms under duality. We consider the torus
The Cremona isomorphism \(\mathbb {T}^E\cong \mathbb {T}^{E^\vee }\) is defined by
Proposition 3.25
(Duality and cokernels of configuration forms). Let \(W\subseteq \mathbb {K}^E\) be a configuration. Then there is an isomorphism over \(\zeta _E\),
where the indices denote localization (see (3.8)). In particular, this induces an isomorphism
Proof
Consider the short exact sequence
and its \(\mathbb {K}\)dual
We identify \(\mathbb {K}^E=\mathbb {K}^{E^{\vee \vee }}\) and \(\mathbb {K}^E/W=W^{\perp \vee }\), and we abbreviate
Then \(Q_{x^E}\) and \(Q^\vee _{x^{E^\vee }}\) are mutual inverses under \(\zeta _E\). Together with (3.22) and (3.23) tensored by \(\mathbb {K}[x^{\pm 1}]\) and (3.20) for W and \(W^\perp \), they fit into a commutative diagram with exact rows connected vertically by morphisms over \(\zeta _E\)
where \([x^{\pm 1}]\) means \(\otimes \mathbb {K}[x^{\pm 1}]\). Exactness of the columns is due to \(\det Q_W=\psi _W\ne 0\) (see Lemma 3.23 and Remark 3.5). Composing the middle vertical isomorphism over \(\zeta _E\) with (taking preimages along) the dashed compositions yields the claimed isomorphism by a diagram chase. \(\square \)
The following result describes the behavior of submaximal minors of configuration forms under deletion–contraction. It is the basis for our inductive approach to second degeneracy schemes.
Lemma 3.26
(Deletion–contraction for submaximal minors). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\) of rank \(r={{\,\mathrm{rk}\,}}\mathsf {M}\), and let \(e\in E\). Then any basis of W/e can be extended to bases of W and \(W{\setminus } e\) such that \(Q_W^{i,j}=\)
for all \(i,j\in {\left\{ 1,\dots ,r\right\} }\). In particular, the \(Q_W^{i,j}\) are linear combinations of squarefree monomials for any basis of W.
Proof
Pick a basis \(w^1,\dots ,w^r\) of \(W\subseteq \mathbb {K}^E\) and consider
as a matrix (see Remark 3.21). Recall that (see Definition 2.17.(b) and (c)),
and the description of (co)loops in Remark 2.18.(a):

If e is a loop, then \(w^i_e=0\) for all \(i=1,\dots ,r\) and hence \(W{\setminus } e=W=W/e\).

If e is not a loop, then we may adjust \(w^1,\dots ,w^r\) such that \(w^i_e=\delta _{i,r}\) for all \(i=1,\dots ,r\) and then \(w^1,\dots ,w^{r1}\) is a general basis of W/e.

If e is a coloop, then we may adjust \(w^r=e\) and \(\pi _{E{\setminus }{\left\{ e\right\} }}\) identifies \(w^1,\dots ,w^{r1}\) with a basis of \(W{\setminus } e=W/e\).
In the latter case,
and the claimed equalities follow (see Lemma 3.23).
It remains to consider the case in which e is not a (co)loop. Then \(\iota _{E{\setminus }{\left\{ e\right\} }}\) and \(\pi _{E{\setminus }{\left\{ e\right\} }}\) (see (2.16)) identify \(w^1,\dots ,w^{r1}\) and \(w^1,\dots ,w^r\) with bases of W/e and \(W{\setminus } e\), respectively. Hence,
where both the entry a and column b are independent of \(x_e\). We consider two cases. If \(i=r\) or \(j=r\), then clearly \(Q_W^{i,j}=Q_{W{\setminus } e}^{i,j}\). Otherwise,
This proves the claimed equalities also in this case (see Lemma 3.23) and the particular claim follows. \(\square \)
As an application of Lemma 3.23, we describe the behavior of configuration polynomials under 2separations.
Proposition 3.27
(Configuration polynomials and 2separations). Let \(W\subseteq \mathbb {K}^E\) be a realization of a connected matroid \(\mathsf {M}\). Suppose that \(E=E_1\sqcup E_2\) is an (exact) 2separation of \(\mathsf {M}\). Then
Proof
We adopt the notation from [30, §8.2]. Extend a basis \(B_2\in \mathcal {B}_{\mathsf {M}\vert _{E_2}}\) to a basis \(B\in \mathcal {B}_\mathsf {M}\). Then W is the row span of a matrix (see [30, (8.1.1)] and Remark 2.15)
where the block columns are indexed by \(B{\setminus } B_2,B_2,E_1{\setminus } B,E_2\setminus B_2\), and \({{\,\mathrm{rk}\,}}D=1\). After suitably ordering and scaling \(B_2\), \(E_1{\setminus } B\) the lower rows of A, we may assume that
The size of b and \(a_1\) is determined by number of rows and columns of D, respectively. While b could be 0, at least one entry of \(a_1\) is a 1. After suitable row operations and adjusting signs of \(B_2\), we can repartition
Denote by \(e\in E\) the index of the column \((0\ 1\ b)^t\). Let \(X_1,x_e,X_2,X_1',X_2'\) be diagonal matrices of variables corresponding to the block columns of A. Then the configuration form of W becomes (see Remark 3.21)
which involves
Laplace expansion of \(\psi _W=\det Q_W\) (see Lemma 3.23) along the eth column yields the claimed formula. \(\square \)
Remark 3.28
(Configuration polynomials and handles). Let \(W\subseteq \mathbb {K}^E\) be a realization of a connected matroid \(\mathsf {M}\), and let \(H\in \mathcal {H}_\mathsf {M}\) be a separating handle. By Lemma 2.4.(e), H is a 2separation of \(\mathsf {M}\). Proposition 3.27 applied to \(E=(E{\setminus } H)\sqcup H\) thus yields the statement of Corollary 3.13 in this case.
Configuration hypersurfaces
In this section, we establish our main results on Jacobian and second degeneracy schemes of realizations of connected matroids: the second degeneracy scheme is Cohen–Macaulay, the Jacobian scheme equidimensional, of codimension 3 (see Theorem 4.25). The second degeneracy scheme is reduced, the Jacobian scheme generically reduced if \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\) (see Theorem 4.25).
Commutative ring basics
In this subsection, we review the relevant preliminaries on equidimensionality and graded Cohen–Macaulayness using the books of Matsumura (see [24]) and Bruns and Herzog (see [7]) as comprehensive references. For the benefit of the nonexperts we provide detailed proofs. Further we relate generic reducedness for a ring and an associated graded ring (see Lemma 4.7).
Equidimensionality of rings
Let R be a Noetherian ring. We denote by \({{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\) and \({{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R\) the sets of minimal and maximal elements of the set \({{\,\mathrm{Spec}\,}}R\) of prime ideals of R with respect to inclusion. The subset \({{\,\mathrm{Ass}\,}}R\subseteq {{\,\mathrm{Spec}\,}}R\) of associated primes of R is finite and \({{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\subseteq {{\,\mathrm{Ass}\,}}R\) (see [24, Thm. 6.5]).
One says that R is catenary if every saturated chain of prime ideals joining \(\mathfrak {p},\mathfrak {q}\in {{\,\mathrm{Spec}\,}}R\) with \(\mathfrak {p}\subseteq \mathfrak {q}\) has (maximal) length \({{\,\mathrm{height}\,}}(\mathfrak {q}/\mathfrak {p})\) (see [24, 31]). We say that R is equidimensional if it is catenary and
If R is a finitely generated \(\mathbb {K}\)algebra, then these two conditions reduce to (see [7, Thm. 2.1.12] and [24, Thm. 5.6])
We say that R is puredimensional if
which implies in particular that \({{\,\mathrm{Ass}\,}}R={{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\). It follows that puredimensional finitely generated \(\mathbb {K}\)algebras are equidimensional.
The following lemma applies to any equidimensional finitely generated \(\mathbb {K}\)algebra.
Lemma 4.1
(Height bound for adding elements). Let R be a Noetherian ring such that \(R_\mathfrak {m}\) is equidimensional for all \(\mathfrak {m}\in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R\).

(a)
All saturated chains of primes in \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}R\) have length \({{\,\mathrm{height}\,}}\mathfrak {p}\).

(b)
For any \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}R\), \(x\in R\) and \(\mathfrak {q}\in {{\,\mathrm{Spec}\,}}R\) minimal over \(\mathfrak {p}+{\left\langle x\right\rangle }\),
$$\begin{aligned} {{\,\mathrm{height}\,}}\mathfrak {q}\le {{\,\mathrm{height}\,}}\mathfrak {p}+1. \end{aligned}$$
Proof

(a)
Take two such chains of length n and \(n'\) starting at minimal primes \(\mathfrak {p}_0\) and \(\mathfrak {p}_0'\), respectively. Extend both by a saturated chain of primes of length m containing \(\mathfrak {p}\) and ending in a maximal ideal \(\mathfrak {m}\). Since \(R_\mathfrak {m}\) is equidimensional by hypothesis, these extended chains have length \(n+m=n'+m\). Therefore, the two chains have length \(n=n'\).

(b)
By Krull’s principal ideal theorem, \({{\,\mathrm{height}\,}}(\mathfrak {q}/\mathfrak {p})\le 1\). Take a chain of primes in \(\mathfrak {p}\) of length \({{\,\mathrm{height}\,}}\mathfrak {p}\) and extend it by \(\mathfrak {q}\) if \(\mathfrak {p}\ne \mathfrak {q}\). By (a), this extended chain has length \({{\,\mathrm{height}\,}}\mathfrak {q}\) and the claim follows. \(\square \)
Lemma 4.2
(Equidimensional finitely generated algebras and localization). Let R be an equidimensional finitely generated \(\mathbb {K}\)algebra and \(x\in R\). If \(R_x\ne 0\), then \(R_x\) is equidimensional of dimension \(\dim R_x=\dim R\).
Proof
Any minimal prime ideal of \(R_x\) is of the form \(\mathfrak {p}_x\) where \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\) with \(x\not \in \mathfrak {p}\). By the Hilbert Nullstellensatz (see [24, Thm. 5.5]),
This yields an \(\mathfrak {m}\in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R\) such that \(\mathfrak {p}\subseteq \mathfrak {m}\not \ni x\) and hence \(\mathfrak {p}_x\subseteq \mathfrak {m}_x\in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R_x\). Since R and hence \(R_x\) is a finitely generated \(\mathbb {K}\)algebra,
by equidimensionality of R. The claim follows. \(\square \)
Generic reducedness
The following types of Artinian local rings coincide: field, regular ring, integral domain and reduced ring (see [24, Thms. 2.2, 14.3]). A Noetherian ring R is generically reduced if the Artinian local ring \(R_\mathfrak {p}\) is reduced for all \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\) (see [24, Exc. 5.2]). This is equivalent to R satisfying Serre’s condition (\(R_0\)). We use the same notions for the associated affine scheme \({{\,\mathrm{Spec}\,}}R\).
Definition 4.3
(Generic reducedness). We call a Noetherian scheme X generically reduced along a subscheme Y if X is reduced at all generic points specializing to a point of Y. If \(X={{\,\mathrm{Spec}\,}}R\) is an affine scheme, then we use the same notions for the Noetherian ring R.
Lemma 4.4
(Reducedness and purity). A Noetherian ring R is reduced if it is generically reduced and puredimensional.
Proof
Since R is puredimensional, \({{\,\mathrm{Ass}\,}}R={{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\), and hence, R becomes a subring of localizations (see [24, Thm. 6.1.(i)])
The latter ring is reduced since R is generically reduced, and the claim follows. \(\square \)
Lemma 4.5
(Reducedness and reduction). Let \((R,\mathfrak {m})\) be a local Noetherian ring. Suppose that R/tR is reduced for a system of parameters t. Then R is regular and, in particular, an integral domain and reduced.
Proof
By hypothesis, R/tR is local Artinian with maximal ideal \(\mathfrak {m}/tR\). Reducedness makes R/tR a field, and hence, \(\mathfrak {m}=tR\). By definition, this means that R is regular. In particular, R is an integral domain and reduced (see [24, Thm. 14.3]). \(\square \)
Definition 4.6
(Rees algebras). Let R be a ring and \(I\unlhd R\) an ideal. The (extended) Rees algebra is the R[t]algebra (see [20, Def. 5.1.1])
The associated graded algebra is the R/Ialgebra
Lemma 4.7
(Generic reducedness from associated graded ring). Let R be a Noetherian ddimensional ring, \(I\unlhd R\) an ideal, \(S:={{\,\mathrm{Rees}\,}}_IR\) and \(\bar{R}:={{\,\mathrm{gr}\,}}_IR\).

(a)
Suppose R is an equidimensional finitely generated \(\mathbb {K}\)algebra. Then S is a \((d+1)\)equidimensional finitely generated \(\mathbb {K}\)algebra.

(b)
If S is \((d+1)\)equidimensional and \(I\ne R\), then \(\bar{R}\) is dequidimensional.

(c)
If S is equidimensional and \(\bar{R}\) is generically reduced, then R is generically reduced along V(I).
Proof
There are ring homomorphisms
Since R is Noetherian, I is finitely generated and S finite type over R.

(a)
If R is an integral domain, then so are \(S\subseteq R[t^{\pm 1}]\). By definition, formation of the Rees ring commutes with base change. After base change to \(R/\mathfrak {p}\) for some \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\), we may assume that R is a ddimensional integral domain. Then S is a \((d+1)\)dimensional integral domain (see [20, Thm. 5.1.4]). Since S is a finitely generated \(\mathbb {K}\)algebra (as R is one), S is equidimensional.

(b)
Multiplication by t is injective on \(R[t^{\pm 1}]\) and hence on S. If \(I\ne R\), then \(S/tS\cong \bar{R}\ne 0\) and t is an Ssequence. Since S is \((d+1)\)equidimensional, \(\bar{R}\) is dequidimensional by Krull’s principal ideal theorem.

(c)
Let \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\) and consider the extension \(\mathfrak {p}[t^{\pm 1}]\in {{\,\mathrm{Spec}\,}}R[t^{\pm 1}]\). Then (see [20, p. 96])
$$\begin{aligned} t\not \in \tilde{\mathfrak {p}}:=\mathfrak {p}[t^{\pm 1}]\cap S\in {{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}S \end{aligned}$$and hence
$$\begin{aligned} S_{\tilde{\mathfrak {p}}}=(S_t)_{\tilde{\mathfrak {p}}_t}=R[t^{\pm 1}]_{\mathfrak {p}[t^{\pm 1}]}. \end{aligned}$$(4.1)Since \(\mathfrak {p}[t^{\pm 1}]\cap R=\mathfrak {p}\), the map \(R\rightarrow R[t^{\pm 1}]\) induces an injection
$$\begin{aligned} R_\mathfrak {p}\hookrightarrow R[t^{\pm 1}]_{\mathfrak {p}[t^{\pm 1}]}. \end{aligned}$$(4.2)To check injectivity, consider \(R_\mathfrak {p}\ni x/1\mapsto 0\in R[t^{\pm 1}]_{\mathfrak {p}[t^{\pm 1}]}\). Then \(0=xy\in R[t^{\pm 1}]\) for some \(y=\sum _iy_it^i\in R[t^{\pm 1}]{\setminus }\mathfrak {p}[t^{\pm 1}]\). Then \(0=xy_i\in R\) for all i and \(y_j\in R{\setminus }\mathfrak {p}\) for some j. It follows that \(0=x/1\in R_\mathfrak {p}\). Combining (4.1) and (4.2) reducedness of \(R_\mathfrak {p}\) follows from reducedness of \(S_{\tilde{\mathfrak {p}}}\).
Suppose now that \(V(\mathfrak {p})\cap V(I)\ne \emptyset \) and hence (the subscript denoting graded parts)
$$\begin{aligned} R\ne \mathfrak {p}+I=\tilde{\mathfrak {p}}_0+(tS)_0=(\tilde{\mathfrak {p}}+tS)_0 \end{aligned}$$implies that \(\tilde{\mathfrak {p}}+tS\ne S\). Let \(\mathfrak {q}\in {{\,\mathrm{Spec}\,}}S\) be a minimal prime ideal over \(\tilde{\mathfrak {p}}+tS\). No minimal prime ideal of S contains the Ssequence \(t\in \mathfrak {q}\). By Lemma 4.1.(b), \({{\,\mathrm{height}\,}}\mathfrak {q}=1\) and \(\mathfrak {q}\) is minimal over t. This makes t a parameter of the localization \(S_\mathfrak {q}\). Under \(S/tS\cong \bar{R}\), the minimal prime ideal \(\mathfrak {q}/tS\in {{\,\mathrm{Spec}\,}}(S/tS)\) corresponds to a minimal prime ideal \(\bar{\mathfrak {q}}\in {{\,\mathrm{Spec}\,}}\bar{R}\). Suppose that \(\bar{R}\) is generically reduced. Then
$$\begin{aligned} S_\mathfrak {q}/tS_\mathfrak {q}=(S/tS)_{\mathfrak {q}/tS}\cong \bar{R}_{\bar{\mathfrak {q}}} \end{aligned}$$is reduced. By Lemma 4.5, \(S_\mathfrak {q}\) and hence its localization \((S_\mathfrak {q})_{\tilde{\mathfrak {p}}_\mathfrak {q}}=S_{\tilde{\mathfrak {p}}}\) is reduced. Then also \(R_\mathfrak {p}\) is reduced, as shown before. \(\square \)
Graded Cohen–Macaulay rings
Let \((R,\mathfrak {m})\) be a Noetherian \(^*\)local ring (see [7, Def. 1.5.13]). By definition, this means that R is a graded ring with unique maximal graded ideal \(\mathfrak {m}\). For any \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}R\), denote by \(\mathfrak {p}^*\in {{\,\mathrm{Spec}\,}}R\) the maximal graded ideal contained in \(\mathfrak {p}\) (see [7, Lem. 1.5.6.(a)]). For any \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}R\), there is a chain of maximal length of graded prime ideals strictly contained in \(\mathfrak {p}\) (see [7, Lem. 1.5.8]). If \(\mathfrak {m}\not \in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R\), then such a chain for \(\mathfrak {n}\in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R\) ends with \(\mathfrak {m}\subsetneq \mathfrak {n}\). It follows that
For any proper graded ideal \(I\lhd R\) also \((R/I,\mathfrak {m}/I)\) is \(^*\)local and
Any associated prime \(\mathfrak {p}\in {{\,\mathrm{Ass}\,}}R\) is graded (see [7, Lem. 1.5.6.(b).(ii)]) and hence \(\mathfrak {p}\subseteq \mathfrak {m}\). This yields a bijection (see [24, Thm. 6.2])
If \(I\unlhd R\) is a graded ideal and \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}R\) minimal over I, then \(\mathfrak {p}/I\in {{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}(R/I)\subseteq {{\,\mathrm{Ass}\,}}(R/I)\), and hence, \(\mathfrak {p}\) is graded.
The following lemma shows in particular that \(^*\)local Cohen–Macaulay rings are pure and equidimensional.
Lemma 4.8
(Height and codimension). Let \((R,\mathfrak {m})\) be a \(^*\)local Cohen–Macaulay ring and \(I\unlhd R\) a graded ideal. Then R is puredimensional and
In particular, R/I is equidimensional if and only if \({{\,\mathrm{height}\,}}\mathfrak {p}={{\,\mathrm{codim}\,}}I\) for all minimal \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}R\) over I.
Proof
The \(^*\)local ring \((R,\mathfrak {m})\) is Cohen–Macaulay if and only if the localization \(R_\mathfrak {m}\) is Cohen–Macaulay (see [7, Exc. 2.1.27.(c)]). In particular, \(R_\mathfrak {m}\) is puredimensional (see [7, Prop. 1.2.13]) and (see [7, Cor. 2.1.4])
Using (4.3), (4.4) for \(I=\mathfrak {p}\) and bijection (4.5), it follows that R is puredimensional:
Using (4.3) and (4.4), (4.6) follows from (4.7):
Since R is Cohen–Macaulay, it is (universally) catenary (see [7, Thm. 2.1.12]). By (4.4) and the preceding discussion of chains of prime ideals in R/I and \(R/\mathfrak {p}\), I is equidimensional if and only if \(\dim (R/I)=\dim (R/\mathfrak {p})\) for all prime ideals \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}R\) minimal over I. The particular claim then follows by (4.6) for I and \(\mathfrak {p}\). \(\square \)
Jacobian and degeneracy schemes
In this subsection, we associate Jacobian and second degeneracy schemes to a configuration. By results of Patterson and Kutz, their supports coincide and their codimension is at most 3.
For a Noetherian ring R, we consider the associated affine (Noetherian) scheme \({{\,\mathrm{Spec}\,}}R\), whose underlying set consists of all prime ideals of R. We refer to elements of \({{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\) as generic points, of \({{\,\mathrm{Ass}\,}}R\) as associated points, and of \({{\,\mathrm{Ass}\,}}R{\setminus }{{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\) as embedded points of \({{\,\mathrm{Spec}\,}}R\). An ideal \(I\unlhd R\) defines a subscheme \({{\,\mathrm{Spec}\,}}(R/I)\subseteq {{\,\mathrm{Spec}\,}}R\).
By abuse of notation we identify
Due to Lemma 4.8,
for any graded ideal \(I\unlhd \mathbb {K}[x]\).
Definition 4.9
Let \(W\subseteq \mathbb {K}^E\) be a configuration. Then the subscheme
is called the configuration hypersurface of W. In particular, \(X_G:=X_{W_G}\) is the graph hypersurface of G (see Definition 2.23). The ideal
is the Jacobian ideal of \(\psi _W\). We call the subschemes (see Definition 3.20)
the Jacobian scheme of \(X_W\) and the second degeneracy scheme of \(Q_W\).
Remark 4.10
(Degeneracy and nonsmooth loci). If \({{\,\mathrm{ch}\,}}\mathbb {K}\not \mid {{\,\mathrm{rk}\,}}\mathsf {M}=\deg \psi \) (see Remark 3.5), then \(\psi _W\) is a redundant generator of \(J_W\) due to the Euler identity. By Lemma 3.23, \(X_W^\text {red}\) and \(\Delta _W^\text {red}\) are the first and second degeneracy loci of \(Q_W\) (see Definition 3.20), whereas \(\Sigma _W^\text {red}\) is the nonsmooth locus of \(X_W\) over \(\mathbb {K}\) (see [24, Thm. 30.3.(1)]). If \(\mathbb {K}\) is perfect, then \(\Sigma _W^\text {red}\) is the singular locus of \(X_W\) (see [24, §28, Lem. 1]).
Remark 4.11
(Loops and line factors). Let \(W\subseteq \mathbb {K}^E\) be a realization of matroid \(\mathsf {M}\). Suppose that e is a loop in \(\mathsf {M}\), that is, \(e^\vee \vert _W=0\). Then \(\psi _W\) and \(Q_W\) are independent of \(x_e\) (see Remark 3.5 and Definition 3.20)
\(\square \)
Lemma 4.12
(Inclusions of schemes). For any configuration \(W\subseteq \mathbb {K}^E\), there are inclusions of schemes \(\Delta _W\subseteq \Sigma _W\subseteq X_W\subseteq \mathbb {K}^E\).
Proof
By definition, \(\psi _W\in J_W\) and hence the second inclusion. By Lemma 3.23, \(\psi _W=\det Q_W\in M_W\) and hence \(\partial _e\psi _W\in M_W\) for all \(e\in E\). Thus, \(J_W\subseteq M_W\) and the first inclusion follows. \(\square \)
Remark 4.13
(Schemes for matroids of small rank). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\).

(a)
If \({{\,\mathrm{rk}\,}}\mathsf {M}\le 1\), then \(\psi _W=1\) (see Remark 3.5) or \(\psi _W\ne 0\) is a \(\mathbb {K}\)linear form. In both cases, \(\Sigma _W=\emptyset =\Delta _W\). If \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\), then \({\left\langle x\right\rangle }\in \Sigma _W\ne \emptyset \ne \Delta _W\ni {\left\langle x\right\rangle }\).

(b)
If \({{\,\mathrm{rk}\,}}\mathsf {M}=2\), then \(\Delta _W\) is a \(\mathbb {K}\)linear subspace of \(\mathbb {K}^E\) and hence an integral scheme. If \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\), the same holds for \(\Sigma _W\) due to the Euler identity (see Remark 4.10). Otherwise, the nonredundant quadratic generator \(\psi _W\) of \(J_W\) can make \(\Sigma _W\) nonreduced (see Example 4.14). \(\square \)
Example 4.14
(Schemes for the triangle). Let \(\mathsf {M}\) be a matroid on \(E\in \mathcal {C}_\mathsf {M}\) with \({\left E\right }=3\) and hence \({{\,\mathrm{rk}\,}}\mathsf {M}={\left E\right }1=2\). Up to scaling and ordering \(E={\left\{ e_1,e_2,e_3\right\} }\), any realization \(W\subseteq \mathbb {K}^E\) of \(\mathsf {M}\) has the basis
With respect to this basis, we compute
It follows that \(\Delta _W\) is a reduced point.
On the other hand,
The matrix expressing the linear generators \(x_1+x_2,x_1+x_3,x_2+x_3\) in terms of the variables \(x_1,x_2,x_3\) has determinant 2. If \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\), then \(J_W={\left\langle x_1,x_2,x_3\right\rangle }\) and \(\Sigma _W\) is a reduced point. Otherwise,
and \(\Sigma _W\) is a nonreduced point.
Lemma 4.15
Consider two sets of variables \(x=x_1,\dots ,x_n\) and \(y=y_1,\dots ,y_m\). Let \(0\ne f\in I\unlhd \mathbb {K}[x]\) and \(0\ne g\in J\unlhd \mathbb {K}[y]\). Then
Proof
For the nonobvious inclusion, take \(h=af+bg\in I[y]\cap J[x]\). Since \(f\in I[y]\), \(bg\in I[y]\) and similarly \(af\in J[x]\). Since \(f\ne 0\) and J are in different variables, it follows that \(a\in J[x]\) and similarly \(b\in I[y]\). \(\square \)
Theorem 4.16
(Decompositions of schemes). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\) without loops. Suppose that \(\mathsf {M}=\bigoplus _{i=1}^n\mathsf {M}_i\) decomposes into connected components \(\mathsf {M}_i\) on \(E_i\). Let \(W=\bigoplus _{i=1}^nW_i\) be the induced decomposition into \(W_i\subseteq \mathbb {K}^{E_i}\) (see Lemma 2.19). Then \(X_W\) is the reduced union of integral schemes \(X_{W_i}\times \mathbb {K}^{E{\setminus } E_i}\), and \(\Sigma _W\) is the union of \(\Sigma _{W_i}\times \mathbb {K}^{E{\setminus } E_i}\) and integral schemes \(X_{W_i}\times X_{W_j}\times \mathbb {K}^{E{\setminus }(E_i\cup E_j)}\) for \(i\ne j\). The same holds for \(\Sigma \) replaced by \(\Delta \). In particular, \(X_W\) is generically smooth over \(\mathbb {K}\).
Proof
Proposition 3.8 yields the claim on \(X_W\) (see Remark 3.5). For the claims on \(\Sigma _W\) and \(\Delta _W\), we may assume that \(n=2\) with \(\mathsf {M}_1\) possibly disconnected. The general case then follows by induction on n.
By Proposition 3.8 and Definition 3.20, \(\psi _W=\psi _{W_1}\cdot \psi _{W_2}\) and \(Q_W=Q_{W_1}\oplus Q_{W_2}\). Then Lemma 4.15 yields
and hence,
The same holds for J and \(\Sigma \) replaced by M and \(\Delta \), respectively.
Suppose now that \(\mathsf {M}\) is connected. By Proposition 3.12, \(\psi _W\not \mid \partial _e\psi _W\) for any \(e\in E\) and hence \(\Sigma _W\subsetneq X_W\). The particular claim follows. \(\square \)
Patterson proved the following result (see [27, Thm. 4.1]). While Patterson assumes \({{\,\mathrm{ch}\,}}\mathbb {K}=0\) and excludes the generator \(\psi _W\in J_W\), his proof works in general (see Remark 4.10). We give an alternative proof using Dodgson identities.
Theorem 4.17
(Nonsmooth loci and second degeneracy schemes). Let \(W\subseteq \mathbb {K}^E\) be a configuration. Then there is an equality of reduced loci
In particular, \(\Sigma _W\) and \(\Delta _W\) have the same generic points, that is,
Proof
Order \(E={\left\{ e_1,\dots ,e_n\right\} }\) and pick a basis \(w=(w^1,\dots ,w^r)\) of W. We may assume that its coefficients with respect to \(e_1,\dots ,e_r\) form an identity matrix, that is, \(w^i_{e_j}=\delta _{i,j}\) for \(i,j\in {\left\{ 1,\dots ,r\right\} }\). For \(i,j\in {\left\{ 1,\dots ,r\right\} }\) denote by \(Q_W^{{\left\{ i,j\right\} },{\left\{ i,j\right\} }}\) the minor of \(Q_W\) obtained by deleting rows and columns i, j. Then there are Dodgson identities (see Remark 3.21, Lemma 3.23 and [6, Lem. 8.2])
for \(i,j\in {\left\{ 1,\dots ,r\right\} }\). In particular, any prime ideal \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}\mathbb {K}[x]\) over \(J_W\) contains \(M_W\) and hence \(\Sigma _W^\text {red}\subseteq \Delta _W^\text {red}\). The opposite inclusion is due to Lemma 4.12. \(\square \)
Corollary 4.18
(Cremona isomorphism). Let \(W\subseteq \mathbb {K}^E\) be a configuration. Then the Cremona isomorphism \(\mathbb {T}^E\cong \mathbb {T}^{E^\vee }\) identifies
In particular, \(\Sigma _W\), \(\Delta _W\), \(\Sigma _{W^\perp }\) and \(\Delta _{W^\perp }\) have the same generic points in \(\mathbb {T}^E\cong \mathbb {T}^{E^\vee }\).
Proof
Propositions 3.10 and 3.25 yield the statements for \(X_W\) and \(\Delta _W\). The statement for \(\Sigma _W\) follows using that \(\zeta _E\) (see (3.21)) identifies \(x_e\partial _e=x_{e^\vee }\partial _{e^\vee }\) for \(e\in E\). The particular claim follows with Theorem 4.17. \(\square \)
Proposition 4.19
(Codimension bound). Let \(W\subseteq \mathbb {K}^E\) be a configuration. Then the codimensions of \(\Sigma _W\) and \(\Delta _W\) in \(\mathbb {K}^E\) are bounded by
In case of equality, \(\Delta _W\) is Cohen–Macaulay (and hence puredimensional) and \(\Sigma _W\) is equidimensional.
Proof
The equality of codimensions follows from Theorem 4.17. The scheme \(\Delta _W\) is defined by the ideal \(M_W\) of submaximal minors of the symmetric matrix \(Q_W\) with entries in the Cohen–Macaulay ring \(\mathbb {K}[x]\) (see [7, 2.1.9]). In particular, \({{\,\mathrm{codim}\,}}_{\mathbb {K}^E}\Sigma _W={{\,\mathrm{grade}\,}}M_W\) (see [7, 2.1.2.(b)]). Kutz proved the claimed inequality and that \(M_W\) is a perfect ideal in case of equality (see [22, Thm. 1]). In the latter case, \(\mathbb {K}[x]/M_W=\mathbb {K}[\Delta _W]\) is a Cohen–Macaulay ring (see [7, Thm. 2.1.5.(a)]) and hence puredimensional (see Lemma 4.8). Then \(\Sigma _W\) is equidimensional by Theorem 4.17. \(\square \)
Generic points and codimension
In this subsection, we show that the Jacobian and second degeneracy schemes reach the codimension bound of 3 in case of connected matroids. The statements on codimension and Cohen–Macaulayness in our main result follow. In the process, we obtain a description of the generic points in relation with any nondisconnective handle.
Lemma 4.20
(Primes over the Jacobian ideal and handles). Let \(W\subseteq \mathbb {K}^E\) be a realization of a connected matroid \(\mathsf {M}\), and let \(H\in \mathcal {H}_\mathsf {M}\) be a proper handle.

(a)
For any \(h\in H\), \(x^{H{\setminus }{\left\{ h\right\} }}\cdot \psi _{W{\setminus } H}\in J_W\).

(b)
For any \(e,f\in H\) with \(e\ne f\), \(x^{H{\setminus }{\left\{ e,f\right\} }}\cdot \psi _{W{\setminus } H}\in J_W+{\left\langle x_e,x_f\right\rangle }\).

(c)
For any \(d\in H\) and \(e\in E{\setminus } H\), \(x^{H{\setminus }{\left\{ d\right\} }}\cdot \partial _e\psi _{W{\setminus } H}\in J_W+{\left\langle x_d\right\rangle }\).

(d)
If \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}\mathbb {K}[x]\) with \(J_W\subseteq \mathfrak {p}\not \ni \psi _{W{\setminus } H}\), then \({\left\langle x_e,x_f,x_g\right\rangle }\subseteq \mathfrak {p}\) for some \(e,f,g\in H\) with \(e\ne f\ne g\ne e\).
Proof
By Remark 3.4 and Corollary 3.13, we may assume that
has the form (3.14).

(a)
Using that \(\psi _W\) is a linear combination of squarefree monomials (see Definition 3.2),
$$\begin{aligned} x^{H{\setminus }{\left\{ h\right\} }}\cdot \psi _{W{\setminus } H}=\psi _W\vert _{x_h=0}=\psi _Wx_h\cdot \partial _h\psi _W\in J_W. \end{aligned}$$ 
(b)
This follows from
$$\begin{aligned} J_W\ni \partial _e\psi _W&=\sum _{h\in H}x^{H{\setminus }{\left\{ e,h\right\} }}\cdot \psi _{W{\setminus } H}+x^{H\setminus {\left\{ e\right\} }}\cdot \psi _{W/H}\\&\equiv x^{H{\setminus }{\left\{ e,f\right\} }}\cdot \psi _{W{\setminus } H}\mod {\left\langle x_e,x_f\right\rangle }. \end{aligned}$$ 
(c)
This follows from
$$\begin{aligned} J_W\ni \partial _e\psi _W&=\sum _{h\in H}x^{H{\setminus }{\left\{ h\right\} }}\cdot \partial _e\psi _{W{\setminus } H}+x^H\cdot \partial _e\psi _{W/H}\\&\equiv x^{H{\setminus }{\left\{ d\right\} }}\cdot \partial _e\psi _{W{\setminus } H}\mod {\left\langle x_d\right\rangle }. \end{aligned}$$ 
(d)
By (a), the hypotheses force \(x^{H{\setminus }{\left\{ h\right\} }}\in \mathfrak {p}\) for all \(h\in H\) and hence \({\left\langle x_e,x_f\right\rangle }\subseteq \mathfrak {p}\) for some \(e,f\in H\) with \(e\ne f\). Then \(x^{H{\setminus }{\left\{ e,f\right\} }}\in \mathfrak {p}\) by (b) and the claim follows. \(\square \)
Remark 4.21
(Primes over the Jacobian ideal and 2separations). Let \(W\subseteq \mathbb {K}^E\) be a realization of a connected matroid \(\mathsf {M}\). Suppose that \(E=E_1\sqcup E_2\) is an (exact) 2separation of \(\mathsf {M}\). For \({\left\{ i,j\right\} }={\left\{ 1,2\right\} }\), note that
and hence by Proposition 3.27
Subtracting \(d_i\cdot \psi _W\) from the latter yields \(\psi _{W\vert _{E_j}}\cdot \psi _{W/E_j}\in J_W\), for \(j=1,2\). It follows that, for every prime ideal \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}\mathbb {K}[x]\) over \(J_W\) and every 2separation F of \(\mathsf {M}\), we have \(\psi _{W\vert _F}\in \mathfrak {p}\) or \(\psi _{W/F}\in \mathfrak {p}\).
Lemma 4.22
(Inductive codimension bound). Let \(W\subseteq \mathbb {K}^E\) be a realization of a connected matroid \(\mathsf {M}\), and let \(H\in \mathcal {H}_\mathsf {M}\) be a proper nondisconnective handle. Suppose that \({{\,\mathrm{codim}\,}}_{\mathbb {K}^{E{\setminus } H}}\Sigma _{W{\setminus } H}=3\). Then \(\Sigma _W\) is equidimensional of codimension
with generic points of the following types:

(a)
\(\mathfrak {p}={\left\langle x_e,x_f,x_g\right\rangle }=:\mathfrak {p}_{e,f,g}\) for some \(e,f,g\in H\) with \(e\ne f\ne g\ne e\),

(b)
\(\mathfrak {p}={\left\langle \psi _{W{\setminus } H},x_d,x_h\right\rangle }=:\mathfrak {p}_{H,d,h}\) for some \(d,h\in H\) with \(d\ne h\),

(c)
\(\psi _{W{\setminus } H},\psi _{W/H}\in \mathfrak {p}\not \ni x_h\) for all \(h\in H\).
Proof
Since H is nondisconnective, \(\psi _{W{\setminus } H}\in \mathbb {K}[x_{E{\setminus } H}]\) is irreducible by Proposition 3.8. Since \(d,h\in H\) with \(d\ne h\), \(\mathfrak {p}_{H,d,h}\in {{\,\mathrm{Spec}\,}}\mathbb {K}[x]\) with \({{\,\mathrm{height}\,}}\mathfrak {p}_{H,d,h}=3\). The same holds for \(\mathfrak {p}_{e,f,g}\).
By Lemma 4.8 and the dimension hypothesis, \(J_{W{\setminus } H}\unlhd \mathbb {K}[x_{E{\setminus } H}]\) has height 3. Thus, for any \(d\in H\),
In particular, \(\Sigma _{W{\setminus } H}\ne \emptyset \) and hence \(\Sigma _W\ne \emptyset \) by Remark 4.13.(a).
Let \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}\mathbb {K}[x]\) be any minimal prime ideal over \(J_W\). By Lemma 4.8 and Proposition 4.19, it suffices to show for the equidimensionality that \({{\,\mathrm{height}\,}}\mathfrak {p}\ge 3\). This follows in particular if \(\mathfrak {p}\) contains a prime ideal of type \(\mathfrak {p}_{e,f,g}\) or \(\mathfrak {p}_{H,d,h}\). By Lemma 4.20.(d), the former is the case if \(\psi _{W{\setminus } H}\not \in \mathfrak {p}\). We may thus assume that \(\psi _{W{\setminus } H}\in \mathfrak {p}\). By Lemma 4.20.(c),
for any \(d\in H\) and \(e\in E{\setminus } H\).
First suppose that \(x_d\in \mathfrak {p}\) for some \(d\in H\). If \(x^{H{\setminus }{\left\{ d\right\} }}\in \mathfrak {p}\), then \(\mathfrak {p}\) contains a prime ideal of type \(\mathfrak {p}_{H,d,h}\) for some \(h\in H{\setminus }{\left\{ d\right\} }\). Otherwise, \({\left\langle J_{W{\setminus } H},x_d\right\rangle }\subseteq \mathfrak {p}\) by (4.9) and hence \({{\,\mathrm{height}\,}}\mathfrak {p}\ge 4\) by (4.8) (see Remark 4.23).
Now suppose that \(x_h\not \in \mathfrak {p}\) for all \(h\in H\) and hence \(\psi _{W/H}\in \mathfrak {p}\) by (3.11) and (3.13) in Corollary 3.13. Let \(\mathfrak {q}\in {{\,\mathrm{Spec}\,}}\mathbb {K}[x]\) be any minimal prime ideal over \(\mathfrak {p}+{\left\langle x_d\right\rangle }\). By (4.9), \(\mathfrak {q}\) contains one of the ideals
for some \(h\in H{\setminus }{\left\{ d\right\} }\). By Lemma 2.4.(b) and (e) (see Remark 3.5),
and hence \(\psi _{W{\setminus } H}\not \mid \psi _{W/H}\) and \(\psi _{W/H}\not \in \mathfrak {p}_{H,d,h}\). Thus, both ideals in (4.10) have height at least 4 (see (4.9)) and hence \({{\,\mathrm{height}\,}}\mathfrak {q}\ge 4\). It follows that \({{\,\mathrm{height}\,}}(\mathfrak {p}+{\left\langle x_d\right\rangle })\ge 4\) and then \({{\,\mathrm{height}\,}}\mathfrak {p}\ge 3\) by Lemma 4.1.(b). \(\square \)
Remark 4.23
The case where \({{\,\mathrm{height}\,}}\mathfrak {p}\ge 4\) in the proof of Lemma 4.22 does finally not occur due to the Cohen–Macaulayness of \(\Delta _W\) achieved by the argument (see Proposition 3.8).
Lemma 4.24
(Generic points for circuits). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\) on \(E\in \mathcal {C}_\mathsf {M}\) with \({\left E\right }1={{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\). Then \(\Sigma _W^\text {red}\) is the union of all codimension3 coordinate subspaces of \(\mathbb {K}^E\).
Proof
We apply the strategy of the proof of Lemma 4.22. By Remark 4.13.(4.13), the rank hypothesis implies that \(\Sigma _W\ne \emptyset \). Let \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}\mathbb {K}[x]\) be any minimal prime ideal over \(J_W\). If \(\psi _{W{\setminus } H}\not \in \mathfrak {p}\) for some \(E\ne H\in \mathcal {H}_\mathsf {M}\), then Lemma 4.20.(d) yields \(e,f,g\in H\) with \(e\ne f\ne g\ne e\) such that \({\left\langle x_e,x_f,x_g\right\rangle }\subseteq \mathfrak {p}\). Otherwise, \(\mathfrak {p}\) contains \(x^{E{\setminus } H}=\psi _{W{\setminus } H}\in \mathfrak {p}\) for all \(E\ne H\in \mathcal {H}_\mathsf {M}\) and hence all \(x_e\) where \(e\in E\). (This can only occur if \({\left E\right }=3\).) By Lemma 4.8 and Proposition 4.19, it follows that \(\mathfrak {p}={\left\langle x_e,x_f,x_g\right\rangle }\). By symmetry, all such triples \(e,f,g\in E\) occur (see Example 3.7). \(\square \)
Theorem 4.25
(Cohen–Macaulayness of degeneracy schemes). Let \(W\subseteq \mathbb {K}^E\) be a realization of a connected matroid \(\mathsf {M}\) of rank \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\). Then \(\Delta _W\) is Cohen–Macaulay (and hence puredimensional) and \(\Sigma _W\) is equidimensional, both of codimension 3 in \(\mathbb {K}^E\).
Proof
By Proposition 4.19, it suffices to show that \({{\,\mathrm{codim}\,}}_{\mathbb {K}^E}\Sigma _W=3\). Lemma 2.13 yields a circuit \(C\in \mathcal {C}_\mathsf {M}\) of size \({\left C\right }\ge 3\) and \({{\,\mathrm{codim}\,}}_{\mathbb {K}^C}\Sigma _{W\vert C}=3\) by Lemma 4.24. Proposition 2.8 yields a handle decomposition of \(\mathsf {M}\) of length k with \(F_1=C\). By Lemma 4.22 and induction on k, then also \({{\,\mathrm{codim}\,}}_{\mathbb {K}^E}\Sigma _{W}=3\). \(\square \)
Corollary 4.26
(Types of generic points). Let \(W\subseteq \mathbb {K}^E\) be a realization of a connected matroid \(\mathsf {M}\) of rank \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\), and let \(H\in \mathcal {H}_\mathsf {M}\) be a nondisconnective handle such that \({{\,\mathrm{rk}\,}}(\mathsf {M}{\setminus } H)\ge 2\). Then all generic points of \(\Sigma _W\) and \(\Delta _W\) are of the types listed in Lemma 4.22 with respect to H.
Proof
Applying Theorem 4.25 to the matroid \(\mathsf {M}{\setminus } H\) with realization \(W{\setminus } H\), the claim follows from Lemma 4.22 and Theorem 4.17. \(\square \)
Corollary 4.27
(Generic points for 3connected matroids). Let \(W\subseteq \mathbb {K}^E\) be a realization of a 3connected matroid \(\mathsf {M}\) with \({\left E\right }>3\) if rank \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\). Then all generic points of \(\Sigma _W\) and \(\Delta _W\) lie in \(\mathbb {T}^E\), that is,
Proof
The equality is due to Theorem 4.17. We may assume that \(\Sigma _W\ne \emptyset \) and hence \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\) by Remark 4.13.(a). Let \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}\Sigma _W\) be a generic point of \(\Sigma _W\). For any \(e\in E\), consider the 1handle \(H:={\left\{ e\right\} }\in \mathcal {H}_\mathsf {M}\). By Proposition 2.5 and Lemma 2.4.(e), H is nondisconnective with \({{\,\mathrm{rk}\,}}(\mathsf {M}{\setminus } H)={{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\). Corollary 4.26 forces \(\mathfrak {p}\) to be of type (c) in Lemma 4.22. It follows that \(\mathfrak {p}\in \bigcap _{e\in E}D(x_e)=\mathbb {T}^E\). \(\square \)
Reducedness of degeneracy schemes
In this subsection, we prove the reducedness statement in our main result as outlined in §1.4.
Lemma 4.28
(Generic reducedness for the prism). Let \(W\subseteq \mathbb {K}^E\) be any realization of the prism matroid (see Definition 2.1). Then \(\Delta _W\cap \mathbb {T}^E\) is an integral scheme of codimension 3, defined by 3 linear binomials, each supported in a corresponding handle. If \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\), then also \(\Sigma _W\cap \mathbb {T}^E=\Delta _W\cap \mathbb {T}^E\).
Proof
By Remark 3.22, we may assume that W is the realization from Lemma 2.25. A corresponding matrix of \(Q_W\) is given in Example 3.24. Reducing its entries modulo \(\mathfrak {p}:={\left\langle x_1+x_2,x_3+x_4,x_5+x_6\right\rangle }\) makes all its \(3\times 3\)minors 0. Therefore, \(J_W\subseteq M_W\subseteq \mathfrak {p}\) by Lemma 4.12. Using the minors
one computes that
By symmetry, it follows that \(x_2x_4x_6\cdot \mathfrak {p}\subseteq M_W\) and hence
Using \(\psi _W\) from Example 3.17, one computes that
By symmetry, it follows that \(2\cdot x_2^2x_4^2x_6^2\cdot \mathfrak {p}\subseteq J_W\) and hence
if \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\). \(\square \)
More details on the prism matroid can be found in Example 5.1.
Lemma 4.29
(Reduction and deletion of non(co)loops). Let \(e\in E\) be a non(co)loop in a matroid \(\mathsf {M}\). For any \(I\unlhd \mathbb {K}[x]\) set
Then \(J_{W{\setminus } e}\subseteq \bar{J}_W\) and \(M_{W{\setminus } e}=\bar{M}_W\) for any realization \(W\subseteq \mathbb {K}^E\) of \(\mathsf {M}\).
Proof
This follows from Proposition 3.12 and Lemma 3.26. \(\square \)
Lemma 4.30
(Generic reducedness and deletion of non(co)loops). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\), and let \(e\in E\) be a non(co)loop. Then \(\Sigma _{W{\setminus } e}=\emptyset \) implies \(\Sigma _W=\emptyset \). Suppose that \({{\,\mathrm{Min}\,}}\Sigma _W\subseteq D(x_e)\) and that \(\Sigma _W\) and \(\Sigma _{W{\setminus } e}\) are equidimensional of the same codimension. If \(\Sigma _{W{\setminus } e}\) is generically reduced, then \(\Sigma _W\) is generically reduced. In this case, each \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}\Sigma _W\) defines a nonempty subset \(\gamma (\mathfrak {p})\subseteq {{\,\mathrm{Min}\,}}\Sigma _{W{\setminus } e}\) such that
In particular, \({\left {{\,\mathrm{Min}\,}}\Sigma _W\right }\le {\left {{\,\mathrm{Min}\,}}\Sigma _{W{\setminus } e}\right }\). The same statements hold for \(\Sigma \) replaced by \(\Delta \).
Proof
The subscheme \(\Sigma _W\cap V(x_e)\subseteq \mathbb {K}^{E{\setminus }{\left\{ e\right\} }}\) is defined by the ideal \(\bar{J}_W\) (see Lemma 4.29). By Lemma 4.29 and since \(J_W\) is graded,
which is the first claim.
Let \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}\Sigma _W\) be a generic point of \(\Sigma _W\). Considered as an element of \({{\,\mathrm{Spec}\,}}\mathbb {K}[x]\) it is minimal over \(J_W\). Since \(J_W\) and hence \(\mathfrak {p}\) is graded, \(\mathfrak {p}+{\left\langle x_e\right\rangle }\ne \mathbb {K}[x]\). Let \(\mathfrak {q}\in {{\,\mathrm{Spec}\,}}\mathbb {K}[x]\) be minimal over \(\mathfrak {p}+{\left\langle x_e\right\rangle }\). By Lemma 4.29,
Since \(x_e\not \in \mathfrak {p}\) by hypothesis, Lemma 4.1 shows that
By the dimension hypothesis, Lemma 4.8 and (4.13), it follows that \(\bar{\mathfrak {q}}\) is minimal over both \(J_{W{\setminus } e}\) and \(\bar{J}_W\). The former means that \(\bar{\mathfrak {q}}\in {{\,\mathrm{Min}\,}}\Sigma _{W{\setminus } e}\). The set \(\gamma (\mathfrak {p})\) of all such \(\bar{\mathfrak {q}}\) is nonempty and satisfies condition (4.11).
Denote by \(t\in \mathbb {K}[\Sigma _W]\) the image of \(x_e\). Then \(\mathfrak {q}\not \in {{\,\mathrm{Min}\,}}\mathbb {K}[\Sigma _W]\) by hypothesis and \(\mathfrak {q}\) is minimal over t since \(\bar{\mathfrak {q}}\) is minimal over \(\bar{J}_W\). This makes t is a parameter of the localization
The inclusion (4.13) gives rise to a surjection of local rings
Suppose now that \(\Sigma _{W{\setminus } e}\) is generically reduced. Then \(\mathbb {K}[\Sigma _{W{\setminus } e}]_{\bar{\mathfrak {q}}}\) is a field which makes (4.14) an isomorphism. By Lemma 4.5, R is then an integral domain with unique minimal prime ideal \(\mathfrak {p}_\mathfrak {q}\). Thus, \(\mathbb {K}[\Sigma _W]_\mathfrak {p}=R_{\mathfrak {p}_\mathfrak {q}}\) is reduced and \(\mathfrak {p}\) is uniquely determined by \(\bar{\mathfrak {q}}\). This uniqueness is condition (4.12). The particular claim follows immediately.
The preceding arguments remain valid if \(\Sigma \) and J are replaced by \(\Delta \) and M, respectively: Lemma 4.29 applies in both cases. \(\square \)
Lemma 4.31
(Initial forms and contraction of non(co)loops). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\). Suppose \(E=F\sqcup G\) is partitioned in such a way that \(\mathsf {M}/G\) is obtained from \(\mathsf {M}\) by successively contracting non(co)loops. For any ideal \(J\unlhd \mathbb {K}[x]_{x^G}=\mathbb {K}[x_F,x_G^{\pm 1}]\), denote by \(J^{\inf }\) the ideal generated by the lowest \(x_F\)degree parts of the elements of J. Then \(J_{W/G}[x_G^{\pm 1}]\subseteq (J_W^{\inf })_{x^G}\) and \(M_{W/G}[x_G^{\pm 1}]\subseteq (M_W^{\inf })_{x^G}\).
Proof
We iterate Proposition 3.12 and Lemma 3.26, respectively, to pass from W to W/G by successively contracting non(co)loops \(e\in G\). This yields a basis of W extending a basis \(w^1,\dots ,w^s\) of W/G such that
for all \(f\in F\) and \(i,j\in {\left\{ 1,\dots ,s\right\} }\), where \(p,q_{i,j}\in \mathbb {K}[x]\) are polynomials with no term divisible by \(x^G\). Since \(\psi _W\) and \(Q_W^{i,j}\) are homogeneous linear combinations of squarefree monomials (see Definition 3.2 and Lemma 3.26), \(x^G\cdot \psi _{W/G}\), \(x^G\cdot \partial _f\psi _{W/G}\) and \(x^G\cdot Q_{W/G}^{i,j}\) are the respective lowest \(x_F\)degree parts in (4.15). The claimed inclusions follow. \(\square \)
Lemma 4.32
(Generic reducedness and contraction of non(co)loops). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\). Suppose \(E=F\sqcup G\) is partitioned in such a way that \(\mathsf {M}/G\) is obtained from \(\mathsf {M}\) by successively contracting non(co)loops. Then \(\Sigma _{W/G}=\emptyset \) implies \(\Sigma _W\cap D(x^G)\cap V(x_F)=\emptyset \). Suppose that \(\Sigma _W\cap D(x^G)\) and \(\Sigma _{W/G}\) are equidimensional of the same codimension. If \(\Sigma _{W/G}\) is generically reduced, then \(\Sigma _W\cap D(x^G)\) is generically reduced along \(V(x_F)\). The same statements hold for \(\Sigma \) replaced by \(\Delta \).
Proof
Consider the ideal
R being equidimensional by hypothesis. With notation from Lemma 4.31
Lemma 4.31 then yields the first claim:
The latter equality makes the second claim vacuous.
We may thus assume that \(I\ne R\). Lemma 4.31 yields a surjection
By Lemmas 4.2 and 4.7 and the dimension hypothesis, source and target are equidimensional of the same dimension and hence \(\pi ^{1}\) induces
Suppose now that \(\Sigma _{W/G}\) and hence \(\Sigma _{W/G}\times \mathbb {T}^G\) is generically reduced. For any \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}\bar{R}\), this makes \(\mathbb {K}[\Sigma _{W/G}\times \mathbb {T}^G]_\mathfrak {p}\) a field and due to
also \(\bar{R}_\mathfrak {p}\) is a field. It follows that \(\bar{R}\) is generically reduced. By Lemma 4.7, R is then generically reduced along V(I). This means that \(\Sigma _W\cap D(x^G)\) is generically reduced along \(V(x_F)\).
The preceding arguments remain valid if \(\Sigma \) and J are replaced by \(\Delta \) and M, respectively: Lemma 4.31 applies in both cases. \(\square \)
Lemma 4.33
(Generic reducedness for circuits). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\) on \(E\in \mathcal {C}_\mathsf {M}\) of rank \({{\,\mathrm{rk}\,}}\mathsf {M}={\left E\right }1\ge 2\). Then \(\Delta _W\) is generically reduced. If \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\), then also \(\Sigma _W\) is generically reduced.
Proof
We proceed by induction on \({\left E\right }\). The case \({\left E\right }=3\) is covered by Example 4.14; here we use \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\).
Suppose now that \({\left E\right }>3\). Let \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}\Sigma _W\) be a generic point of \(\Sigma _W\). By Lemma 4.24, \(\mathfrak {p}={\left\langle x_e,x_f,x_g\right\rangle }\) for some \(e,f,g\in H\) with \(e\ne f\ne g\ne e\). Pick \(d\in E{\setminus }{\left\{ e,f,g\right\} }\). Then \(E{\setminus }{\left\{ d\right\} }\in \mathcal {C}_{\mathsf {M}/d}\) and hence \(\Sigma _{W/d}\) is generically reduced by induction. By Lemmas 4.2 and 4.32, \(\Sigma _W\cap D(x_d)\) is then along \(V(x_{E{\setminus }{\left\{ d\right\} }})\). By choice of d, \({\left\langle x_{E{\setminus }{\left\{ d\right\} }}\right\rangle }\in V(\mathfrak {p})\cap D(x_d)\). In other words, \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}(\Sigma _W\cap D(x_d))\) specializes to a point in \(V(x_{E{\setminus }{\left\{ d\right\} }})\cap D(x_d)\). Thus, \(\Sigma _W\) is reduced at \(\mathfrak {p}\). It follows that \(\Sigma _W\) is generically reduced.
By Theorem 4.17, \(\Delta _W\) has the same generic points as \(\Sigma _W\). Therefore, the preceding arguments remain valid if \(\Sigma \) is replaced by \(\Delta \). \(\square \)
Lemma 4.34
(Generic reducedness and contraction of nonmaximal handles). Let \(W\subseteq \mathbb {K}^E\) be a realization of a connected matroid \(\mathsf {M}\) of rank \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\). Assume that \({\left {{\,\mathrm{Max}\,}}\mathcal {H}_\mathsf {M}\right }\ge 2\) and set
Suppose that \(\Sigma _{W'}\) is generically reduced for every realization \(W'\subseteq \mathbb {K}^{E'}\) of every connected matroid \(\mathsf {M}'\) of rank \({{\,\mathrm{rk}\,}}\mathsf {M}'\ge 2\) with \({\left E'\right }<{\left E\right }\).

(a)
If \(\hbar >3\), then \(\Sigma _W\) is generically reduced.

(b)
If \(\hbar >2\) and \(e\in E\), then \(\Sigma _W\) is reduced at all \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}\Sigma _W\cap V(x_e)\).
The same statements hold for \(\Sigma \) replaced by \(\Delta \).
Proof
Let \(\mathfrak {p}\in {{\,\mathrm{Spec}\,}}\mathbb {K}[x]\) with \({{\,\mathrm{height}\,}}\mathfrak {p}=3\). Pick a subset \(F\subseteq E\) such that \({\left F\cap H'\right }=1\) for all \(H'\in {{\,\mathrm{Max}\,}}\mathcal {H}_M\). If possible, pick \(F\cap H'={\left\{ e\right\} }\) such that \(x_e\in \mathfrak {p}\). If \(\hbar >3\), then by Lemma 4.1.(b)
If \(\hbar >2\) and \(\mathfrak {p}\in V(x_e)\), then (4.16) holds with 3 replaced by 2. In either case pick \(\mathfrak {q}\in {{\,\mathrm{Spec}\,}}\mathbb {K}[x]\) such that
Add to F all \(f\in E\) with \(x_f\in \mathfrak {q}\). This does not affect (4.17). Then \(x_g\not \in \mathfrak {q}\) and hence \(x_g\not \in \mathfrak {p}\) for all \(g\in G:=E{\setminus } F\ne \emptyset \). In other words,
By the initial choice of F, \(G\cap H'\subsetneq H'\) for each \(H'\in {{\,\mathrm{Max}\,}}\mathcal {H}_\mathsf {M}\). By Lemma 2.4.(d), successively contracting all elements of G does, up to bijection, not affect circuits and maximal handles. In particular, \(\mathsf {M}/G\) is a connected matroid on the set F, obtained from \(\mathsf {M}\) by successively contracting non(co)loops.
Since \({\left F\right }\ge {\left {{\,\mathrm{Max}\,}}\mathcal {H}_\mathsf {M}\right }\ge 2\), connectedness implies that \({{\,\mathrm{rk}\,}}(\mathsf {M}/G)\ge 1\). If \({{\,\mathrm{rk}\,}}(\mathsf {M}/G)=1\), then \(\Sigma _{W/G}=\emptyset \) by Remark 4.13.(a). Then \(\Sigma _W\cap D(x^G)\cap V(x_F)=\emptyset \) by Lemma 4.32 and hence \(\mathfrak {p}\not \in \Sigma _W\) by (4.18).
Suppose now that \(\mathfrak {p}\in \Sigma _W\) and hence \({{\,\mathrm{rk}\,}}(\mathsf {M}/G)\ge 2\). Then \(\Sigma _{W/G}\) is generically reduced by hypothesis, and \(\mathfrak {p}\in \Sigma _W\cap D(x^G)\) specializes to a point in \(V(x_F)\cap D(x^G)\) by (4.18). By Theorem 4.25 and Lemma 4.2, \(\Sigma _W\), \(\Sigma _W\cap D(x^G)\) and \(\Sigma _{W/G}\) are equidimensional of codimension 3. By Lemma 4.8, \({{\,\mathrm{height}\,}}\mathfrak {p}=3\) means that \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}\Sigma _W\). By Lemma 4.32, \(\Sigma _W\) is thus reduced at \(\mathfrak {p}\). The claims follow.
The preceding arguments remain valid if \(\Sigma \) is replaced by \(\Delta \). \(\square \)
Lemma 4.35
(Reducedness for connected matroids). Let \(W\subseteq \mathbb {K}^E\) be a realization of a connected matroid \(\mathsf {M}\) of rank \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\). Then \(\Delta _W\) is reduced. If \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\), then \(\Sigma _W\) is generically reduced.
Proof
By Theorem 4.25, \(\Delta _W\) is puredimensional. By Lemma 4.4, \(\Delta _W\) is thus reduced if it is generically reduced. By Lemma 4.12 and Theorem 4.17, the first claim follows if \(\Sigma _W\) is generically reduced.
Assume that \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\). We proceed by induction on \({\left E\right }\). By Lemma 4.33, \(\Sigma _W\) is generically reduced if \(E\in \mathcal {C}_\mathsf {M}\); the base case where \({\left E\right }=3\) needs \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\). Otherwise, by Proposition 2.8, \(\mathsf {M}\) has a handle decomposition of length \(k\ge 2\). By Proposition 2.12, \(\mathsf {M}\) has \(k+1\) (disjoint) nondisconnective handles \(H=H_1,\dots ,H_\ell \in \mathcal {H}_\mathsf {M}\) with
Note that \(H_1,\dots ,H_\ell \in {{\,\mathrm{Max}\,}}\mathcal {H}_\mathsf {M}\cap \mathcal {I}_\mathsf {M}\) by Lemma 2.4.(c) and (b). In particular, \({{\,\mathrm{rk}\,}}(\mathsf {M}{\setminus } H)\ne 0\).
Suppose first that \(H={\left\{ h\right\} }\). Then \({{\,\mathrm{rk}\,}}(\mathsf {M}{\setminus } h)\ge 2\) by Remark 4.13.(a) and Lemma 4.30, and \({{\,\mathrm{Min}\,}}\Sigma _W\subseteq D(x_h)\) by Corollary 4.26. By Theorem 4.25, both \(\Sigma _W\) and \(\Sigma _{W{\setminus } h}\) are equidimensional of codimension 3. Thus, \(\Sigma _W\) is generically reduced by Lemma 4.30 and the induction hypothesis.
Suppose now that \({\left H_i\right }\ge 2\) for all \(i=1,\dots ,\ell \), and set (see Lemma 4.34)
If \(\hbar >3\), then \(\Sigma _W\) is generically reduced by Lemma 4.34.(a) and the induction hypothesis. Otherwise,
and hence \(2\ell \le \sum _{i=1}^\ell {\left H_i\right }\le 3+\ell \). Comparing with (4.19) yields \(\ell =3\), \(k=2\) and \({\left H_i\right }=2\) for \(i=1,2,3\). By Lemma 2.10, \(E=H_1\sqcup H_2\sqcup H_3\) is then the handle partition of \(\mathsf {M}\). In particular, \(\hbar =63=3>2\). By Lemma 2.25, \(\mathsf {M}\) must be the prism matroid.
Let now \(\mathfrak {p}\in {{\,\mathrm{Min}\,}}\Sigma _W\) be a generic point of \(\Sigma _W\), with \(\mathsf {M}\) the prism matroid. If \(\mathfrak {p}\in \mathbb {T}^E\), then \(\Sigma _W\) is reduced at \(\mathfrak {p}\) by Lemma 4.28; here we use \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\) again. Otherwise, \(\mathfrak {p}\in V(x_e)\) for some \(e\in E\). Then \(\Sigma _W\) is reduced at \(\mathfrak {p}\) by Lemma 4.34.(b) and the induction hypothesis.
The preceding arguments remain valid for arbitrary \({{\,\mathrm{ch}\,}}\mathbb {K}\) if \(\Sigma \) is replaced by \(\Delta \). \(\square \)
Theorem 4.36
(Reducedness). Let \(W\subseteq \mathbb {K}^E\) be a realization of a matroid \(\mathsf {M}\). Then
is reduced. If \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\), then \(\Sigma _W\) is generically reduced.
Proof
By Theorem 4.16 and Lemma 4.35 (see Remarks 4.11 and 4.13.(a)), \(\Delta _W\) is reduced and \(\Sigma _W\) is generically reduced if \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\). The claimed equality is then due to Theorem 4.17. \(\square \)
Integrality of degeneracy schemes
In this subsection, we prove the following companion result to Proposition 3.8 as outlined in §1.4.
Theorem 4.37
(Integrality for 3connected matroids). Let \(W\subseteq \mathbb {K}^E\) be a realization of a 3connected matroid \(\mathsf {M}\) of rank \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\). Then \(\Delta _W\) is integral and hence \(\Sigma _W\) is irreducible.
Proof
The claim on \(\Delta _W\) follows from Remark 4.13.(a) and Lemmas 4.38 and 4.43 and Corollary 4.41. Theorem 4.17 yields the claim on \(\Sigma _W\). \(\square \)
In the following, we use notation from Example 2.26.
Lemma 4.38
(Reduction to wheels and whirls). It suffices to verify Theorem 4.37 for \(\mathsf {M}\in {\left\{ \mathsf {W}_n,\mathsf {W}^n\right\} }\) with \(n\ge 3\).
Proof
Let \(\mathsf {M}\) and W be as in Theorem 4.37. By Remark 4.13.(b) and Theorem 4.17, the claim holds if \({{\,\mathrm{rk}\,}}\mathsf {M}=2\). If \({\left E\right }\le 4\), then \(\mathsf {M}=\mathsf {U}_{2,n}\) where \(n\in {\left\{ 3,4\right\} }\) (see [26, Tab. 8.1]) and hence \({{\,\mathrm{rk}\,}}\mathsf {M}=2\). We may thus assume that \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 3\) and \({\left E\right }\ge 5\).
The 3connectedness hypothesis on \(\mathsf {M}\) holds equivalently for \(\mathsf {M}^\perp \) (see 2.10). By Corollaries 4.18 and 4.27, the Cremona isomorphism thus identifies
It follows that integrality is equivalent for \(\Delta _W\) and \(\Delta _{W^\perp }\). In particular, we may also assume that \({{\,\mathrm{rk}\,}}\mathsf {M}^\perp \ge 3\).
We proceed by induction on \({\left E\right }\). Suppose that \(\mathsf {M}\) is not a wheel or a whirl. Since \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 3\), Tutte’s wheelsandwhirls theorem (see [26, Thm. 8.8.4]) yields an \(e\in E\) such that \(\mathsf {M}{\setminus } e\) or \(\mathsf {M}/e\) is again 3connected. In the latter case, we replace W by \(W^\perp \) and use (2.11). We may thus assume that \(\mathsf {M}{\setminus } e\) is 3connected. Then \(\Delta _{W{\setminus } e}\) is integral by induction hypothesis. Note that \({{\,\mathrm{Min}\,}}\Delta _W\subseteq D(x_e)\) by (4.20). By Theorem 4.25, \(\Delta _W\) and \(\Delta _{W{\setminus } e}\) are equidimensional of codimension 3. By Remark 4.13.(a) and Lemma 4.30, \(\Delta _W\ne \emptyset \) and \({\left {{\,\mathrm{Min}\,}}\Delta _W\right }\le {\left {{\,\mathrm{Min}\,}}\Delta _{W{\setminus } e}\right }=1\). It follows that \(\Delta _W\) is integral. \(\square \)
Lemma 4.39
(Turning wheels). Let \(W\subseteq \mathbb {K}^E\) be the realization of \(\mathsf {W}_n\) from Lemma 2.27. Then the cyclic group \(\mathbb {Z}_n\) acts on \(X_W\), \(\Sigma _W\) and \(\Delta _W\) by “turning the wheel,” induced by the generator \(1\in \mathbb {Z}_n\) mapping
Proof
By Lemma 2.27, W has a basis \(w=(w_1,\dots ,w_n)\) where \(w^i=s_i+r_ir_{i1}\) for all \(i\in \mathbb {Z}_n\). The assignment (4.21) stabilizes \(W\subseteq \mathbb {K}^E\). The resulting \(\mathbb {Z}_n\)action stabilizes \(\psi _W\) and \(Q_W\), and hence \(J_W\) and \(M_W\). As a consequence, it induces an action on \(X_W\), \(\Sigma _W\) and \(\Delta _W\). \(\square \)
The graph hypersurface of the nwheel was described by Bloch, Esnault and Kreimer (see [6, (11.5)]). We show that it is also the unique configuration hypersurface of the nwhirl.
Proposition 4.40
(Schemes for wheels and whirls). Let \(W\subseteq \mathbb {K}^E\) be any realization of \(\mathsf {M}\in {\left\{ \mathsf {W}_n,\mathsf {W}^n\right\} }\) where \(E=S\sqcup R\). Then there are coordinates \(z'_1,\dots ,z'_n,y_1,\dots ,y_n\) on \(\mathbb {K}^E\) such that
where
In particular, \(X_W\), \(\Sigma _W\) and \(\Delta _W\) depend only on n up to isomorphism.
Proof
We may assume that W is the realization from Lemma 2.27. Denote the coordinates on \(\mathbb {K}^E=\mathbb {K}^{S\sqcup R}\) by
and consider the \(\mathbb {K}\)linear automorphism defined by
Then \(Q_W\) is represented by the matrix
Suitable scaling of \(y_1,\dots ,y_n\) turns this matrix into \(Q_n\). The particular claim follows with Lemma 3.23. \(\square \)
Corollary 4.41
(Small wheels and whirls). Theorem 4.37 holds for the matroids \(\mathsf {M}=\mathsf {W}_3\) and \(\mathsf {M}=\mathsf {W}^n\) for \(n\le 4\).
Proof
Let W be any realization of \(\mathsf {M}\). By Theorem 4.36, \(\Delta _W\) is reduced and it suffices to check irreducibility, replacing \(\mathbb {K}\) by its algebraic closure. By Proposition 4.40, we may assume that \(\Delta _W=V(I_{k+1}(Q_n))\) for \(k=n2\).
Consider the morphism of algebraic varieties of matrices
Let \(y_{i,j}\) and \(z_{i,j}\) be the coordinates on Y and Z, respectively. Then \(\Delta _W\) identifies with \(V(z_{1,3},z_{2,4})\subseteq Z\) for \(n=4\) and with Z itself for \(n\le 3\). Both the preimage Y of Z and for \(n=4\) the preimage
of \(V(z_{1,3},z_{2,4})\) are irreducible. It thus suffices to show that Y surjects onto Z, which holds for all \(k\le n\).
Let \(A\in Z\) and \(I\subseteq {\left\{ 1,\dots ,n\right\} }\) with \({\left I\right }={{\,\mathrm{rk}\,}}A=k\) and rows \(i\in I\) of A linearly independent. Apply row operations C to make the rows \(i\not \in I\) of CA zero. Then \(CAC^t\) is nonzero only in rows and columns \(i\in I\). Modifying C to include further row operations turns \(CAC^t\) into a diagonal matrix. As \(\mathbb {K}\) is algebraically closed, \(CAC^t=D^2\) where D has exactly k nonzero diagonal entries. Then \(A=BB^t\) where \(B:=C^{1}D\), considered as an element of Y by dropping zero columns. \(\square \)
Lemma 4.42
(Operations on wheels and whirls). Let \(\mathsf {M}\in {\left\{ \mathsf {W}_n,\mathsf {W}^n\right\} }\).

(a)
The bijection
$$\begin{aligned} E=S\sqcup R\rightarrow E^\vee ,\quad s_i\mapsto r_i^\vee ,\quad r_i\mapsto s_i^\vee , \end{aligned}$$identifies \(\mathsf {M}=\mathsf {M}^\perp \).
Suppose now that n is not minimal for \(\mathsf {M}\) to be defined, that is, \(n>3\) if \(\mathsf {M}=\mathsf {W}_n\) and \(n>2\) if \(\mathsf {M}=\mathsf {W}^n\).

(b)
The matroid \(\mathsf {M}{\setminus } s_n\) is connected of rank \({{\,\mathrm{rk}\,}}(\mathsf {M}{\setminus } s_n)\ge 2\). Its handle partition consists of nondisconnective handles, the 2handle \({\left\{ r_{n1},r_n\right\} }\) and 1handles.

(c)
The matroid \(\mathsf {M}/r_n\) is connected of rank \({{\,\mathrm{rk}\,}}(\mathsf {M}/r_n)\ge 2\). Its handle partition consists of nondisconnective 1handles.

(d)
We can identify \(\mathsf {W}_n{\setminus } s_n/r_n=\mathsf {W}_{n1}\) and \(\mathsf {W}^n{\setminus } s_n/r_n=\mathsf {W}^{n1}\).
Proof

(a)
The selfduality claim is obvious (see [26, Prop. 8.4.4]).

(b)
This follows from the description of connectedness in terms of circuits (see (2.5) and Example 2.26).

(c)
This follows from the description of connectedness in terms of circuits (see (2.7) and Example 2.26).

(d)
The operation \(\mathsf {M}\mapsto \mathsf {M}{\setminus } s_n/r_n\) deletes the triangle \({\left\{ s_{n1},r_{n1},s_n\right\} }\) and maps the triangle \({\left\{ s_{n},r_{n},s_1\right\} }\) to \({\left\{ s_{n1},r_{n1},s_1\right\} }\) (see (2.5) and (2.7)). By duality, it acts on triads in the same way (see (a) and (2.11)). Moreover, \(R\in \mathcal {C}_{\mathsf {M}{\setminus } s_n/r_n}\) is equivalent to \(R\in \mathcal {C}_\mathsf {M}\) and hence \(\mathsf {M}=\mathsf {W}_n\) (see (2.5), (2.7) and Example 2.26). The claim then follows using the characterization of wheels and whirl in terms of triangles and triads (see Example 2.26). \(\square \)
Lemma 4.43
(Induction on wheels and whirls). Theorem 4.37 for \(\mathsf {M}=\mathsf {W}_n\) and \(\mathsf {M}=\mathsf {W}^n\) follows from the cases \(n=3\) and \(n\le 4\), respectively.
Proof
Suppose that n is not minimal for \(\mathsf {M}\in {\left\{ \mathsf {W}_n,\mathsf {W}^n\right\} }\) to be defined. Let \(W'\) be any realization of \(\mathsf {M}/r_n\). Then \(W'{\setminus } s_n\) is a realization of
by Lemma 4.42.(d). By induction hypothesis and Corollary 4.27, \(\Delta _{W'{\setminus } s_n}\) is integral with generic point in \(\mathbb {T}^{E{\setminus }{\left\{ s_n,r_n\right\} }}\). By Lemma 4.42.(c) and Corollary 4.26, \({{\,\mathrm{Min}\,}}\Delta _{W'}\subseteq \mathbb {T}^{E{\setminus }{\left\{ r_n\right\} }}\subseteq D(s_n)\). By Lemma 4.42.(c) and Theorems 4.25, \(\Delta _{W'}\) and \(\Delta _{W'{\setminus } s_n}\) are equidimensional of codimension 3. By Remark 4.13.(a) and Lemma 4.30, \(\Delta _{W'}\) is then integral.
Let W be any realization of \(\mathsf {M}\) and use the coordinates from (4.22). By Lemma 4.42.(b) and Corollary 4.26, \(\Delta _{W{\setminus } s_n}\) has at most one generic point \(\mathfrak {q}'\) in \(V(y_{n1},y_n)\) while all the others lie in \(\mathbb {T}^{E{\setminus }{\left\{ s_n\right\} }}\). By Corollary 4.18, the Cremona isomorphism identifies the latter with generic points of \(\Delta _{(W{\setminus } s_n)^\perp }\) in \(\mathbb {T}^{E^\vee {\setminus }{\left\{ s_n^\vee \right\} }}\). Use (2.11) and Lemma 4.42.(a) to identify
and consider \((W{\setminus } s_n)^\perp \) as a realization \(W'\) of \(\mathsf {M}/r_n\). By the above, \(\Delta _{W'}\) is integral with generic point in \(\mathbb {T}^{E{\setminus }{\left\{ r_n\right\} }}\). Thus, \(\Delta _{W{\setminus } s_n}\) has a unique generic point \(\mathfrak {q}\) in \(\mathbb {T}^{E{\setminus }{\left\{ s_n\right\} }}\). To summarize,
By Lemma 4.42.(b) and Theorems 4.25 and 4.36, \(\Delta _W\) and \(\Delta _{W{\setminus } s_n}\) are equidimensional of codimension 3 and reduced. It suffices to show that \(\Delta _W\) is irreducible. By way of contradiction, suppose that \(\mathfrak {p}\ne \mathfrak {p}'\) for some \(\mathfrak {p},\mathfrak {p}'\in {{\,\mathrm{Min}\,}}\Delta _W\). By Corollary 4.27, \({{\,\mathrm{Min}\,}}\Delta _W\subseteq \mathbb {T}^E\subseteq D(s_n)\). By Lemma 4.30 and (4.23), it follows that
By (4.11) in Lemma 4.30, we may assume that \(\sqrt{\bar{\mathfrak {p}}}=\mathfrak {q}\) and \(\sqrt{\bar{\mathfrak {p}}'}=\mathfrak {q}'\) where \(\bar{I}:=(I+{\left\langle z_n\right\rangle })/{\left\langle z_n\right\rangle }\).
Consider first the case where \(\mathsf {M}=\mathsf {W}_n\) with \(n\ge 4\). By Remark 3.22, we may assume that W is the realization from Lemma 2.27. By Lemma 4.39, the cyclic group \(\mathbb {Z}_n\) acts on \({\left\{ \mathfrak {p},\mathfrak {p}'\right\} }\) by “turning the wheel.” If it acts identically, then \(\sqrt{\mathfrak {p}'+{\left\langle z_i\right\rangle }}\supseteq {\left\langle y_{i1},y_i\right\rangle }\) for all \(i=1,\dots ,n\) and hence
Then \({{\,\mathrm{height}\,}}(\mathfrak {p}'+{\left\langle z_1,\dots ,z_n\right\rangle })=2n\) which implies \({{\,\mathrm{height}\,}}\mathfrak {p}'\ge n>3\) by Lemma 4.1.(b) , contradicting Theorem 4.25 (see Lemma 4.8). Otherwise, the generator \(1\in \mathbb {Z}_n\) switches the assignment \(\mathfrak {p}\mapsto \mathfrak {q}\) and \(\mathfrak {p}\mapsto \mathfrak {q}'\) and \(n=2m\) must be even. Then \(\sqrt{\mathfrak {p}+{\left\langle z_{2i}\right\rangle }}\supseteq {\left\langle y_{2i1},y_{2i}\right\rangle }\) for all \(i=1,\dots ,m\) and hence
This leads to a contradiction as before.
Consider now the case where \(\mathsf {M}=\mathsf {W}^n\) with \(n\ge 5\). For \(i=1,\dots ,n\), denote by \(\mathfrak {q}_i\) and \(\mathfrak {q}'_i\) the generic points of \(\Delta _{W{\setminus } s_i}\) as in (4.23). By the pigeonhole principle, one of \(\mathfrak {p}\) and \(\mathfrak {p}'\), say \(\mathfrak {p}\), is assigned to \(\mathfrak {q}'_i\) for 3 spokes \(s_i\). In particular, \(\mathfrak {p}\) is assigned to \(\mathfrak {q}'_i\) and \(\mathfrak {q}'_j\) for two nonadjacent spokes \(s_i\) and \(s_j\). Then
This leads to a contradiction as before. The claim follows. \(\square \)
Theorem 4.37 proves the “only if” part of the following conjecture.
Conjecture 4.44
(Irreducibility and 3connectedness). Let \(\mathsf {M}\) be a matroid of rank \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\) on E. Then \(\mathsf {M}\) is 3connected if and only if, for some/any realization \(W\subseteq \mathbb {K}^E\) of \(\mathsf {M}\), both \(\Delta _W\) and \(\Delta _{W^\perp }\) are integral.
Examples
In this section, we illustrate our results with examples of prism, whirl and uniform matroids.
Example 5.1
(Prism matroid). Consider the prism matroid \(\mathsf {M}\) (see Definition 2.1) with its unique realization W (see Lemma 2.25). Then
by Example 3.17. By Lemma 4.28, \(\Delta _W\) has the unique generic point
in \(\mathbb {T}^6\). By Corollary 4.26, there can be at most 3 more generic points symmetric to
Over \(\mathbb {K}=\mathbb {F}_2\), their presence is confirmed by a computation in Singular (see [14]). It reveals a total of 7 embedded points in \(\Sigma _W\). There is \({\left\langle x_1,\dots ,x_6\right\rangle }\), and 3 symmetric to each of
Moreover, \(\Sigma _W\) is not reduced at any generic point. Since the above associated primes are geometrically prime, the conclusions remain valid over any field \(\mathbb {K}\) with \({{\,\mathrm{ch}\,}}\mathbb {K}=2\).
A Singular computation over \(\mathbb {Q}\) shows that \(\Sigma _W\) has exactly the above associated points for any field \(\mathbb {K}\) with \({{\,\mathrm{ch}\,}}\mathbb {K}=0\) or \({{\,\mathrm{ch}\,}}\mathbb {K}\gg 0\). We expect that this holds in fact for \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\).
To verify at least the presence of these associated points in \(\Sigma _W\) for \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\), we claim that
The colon ideals on the right hand side can be read off from a suitable Gröbner basis (see [17, Lems. 1.8.3, 1.8.10 and 1.8.12]). Using Singular we compute such a Gröbner basis over \(\mathbb {Z}\) which confirms our claim. There are no odd prime numbers dividing its leading coefficients. It is therefore a Gröbner basis over any field \(\mathbb {K}\) with \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\) and the argument remains valid.
Example 5.2
(Whirl matroid). Consider the whirl matroid \(\mathsf {M}:=\mathsf {W}^3\) (see Example 2.26). It is realized by 6 points in \(\mathbb {P}^2\) with the collinearities shown in Fig. 4. Since \(\mathsf {M}\) contracts to the uniform matroid \(\mathsf {U}_{2,4}\), \(\mathsf {M}\) is not regular (see [26, Thm. 6.6.6]). The configuration polynomial reflects this fact. Using the realization W of \(\mathsf {M}\) from Lemma 2.27 with \(t=1\), \({\left\{ s_1,s_2,s_3\right\} }={\left\{ 1,2,3\right\} }\) and \({\left\{ r_1,r_2,r_3\right\} }={\left\{ 4,5,6\right\} }\), we find
Replacing in \(\psi _W\) the coefficient 4 of \(x_4x_5x_6\) by a 1 yields the matroid polynomial \(\psi _\mathsf {M}\) (see Remark 3.6).
By Theorem 4.25, the configuration hypersurface \(X_W\) defined by \(\psi _W\) has 3codimensional nonsmooth locus \(\Sigma _W^\text {red}\). Using Singular (see [14]) we compute a Gröbner basis over \(\mathbb {Z}\) of the ideal of partial derivatives of \(\psi _\mathsf {M}\). The only prime numbers dividing leading coefficients are 2, 3 and 5. For \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2,3,5\), it is therefore a Gröbner basis over \(\mathbb {K}\). From its leading exponents we calculate that the nonsmooth locus of the hypersurface defined by \(\psi _\mathsf {M}\) has codimension 4 (see [17, Cor. 5.3.14]). By further Singular computations, this codimension is 4 for \({{\,\mathrm{ch}\,}}\mathbb {K}=2,5\), and 3 for \({{\,\mathrm{ch}\,}}\mathbb {K}=3\).
Example 5.3
(Uniform rank3 matroid). Suppose that \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2,3\). Then the configuration \(W={\left\langle w^1,w^2,w^3\right\rangle }\subseteq \mathbb {K}^3\) defined by
realizes the uniform matroid \(\mathsf {U}_{3,6}\) (see Example 2.20). The entries of \(Q_w=(q_{i,j})_{i,j}\) satisfy the linear dependence relation (see Remark 3.21)
By Lemma 3.23, \(\psi _W\) thus depends on fewer than 6 variables. More precisely, a Singular computation shows that \(\Sigma _W\) has Betti numbers (1, 5, 10, 10, 5, 1), is not reduced and hence not Cohen–Macaulay.
Now, take \(W'\) to be a generic realization of \(\mathsf {U}_{3,6}\). Then the entries of \(Q_{W'}\) with indices (i, j) where \(i\le j\) are linearly independent (see [5, Prop. 6.4]), and \(\Sigma _{W'}\) is reduced Cohen–Macaulay with Betti numbers (1, 6, 8, 3). So basic geometric properties of the configuration hypersurface \(X_W\) are not determined by the matroid \(\mathsf {M}\), but depend on the realization W.
Example 5.4
(Uniform rank2 matroid). Suppose that \({{\,\mathrm{ch}\,}}\mathbb {K}\ne 2\) and consider the uniform matroid \(\mathsf {U}_{2,n}\) for \(n\ge 3\) (see Examples 2.2 and 3.7.(c)). A realization W of \(\mathsf {U}_{2,n}\) is spanned by two vectors \(w^1,w^2\in \mathbb {K}^n\) for which (see Example 2.20)
for \(1\le i<j\le n\). Then
and the ideal \(J_W\) is generated by n linear forms. These forms may be written as the rows of the Hessian matrix
where by convention \(c_{W,{\left\{ i,i\right\} }}=0\). Since uniform matroids are connected, Theorem 4.25 implies that \(H_W\) has rank exactly 3.
For \(n\ge 4\), this amounts to a classicallooking linear algebra fact: suppose that \(A=(a_{i,j}^2)_{i,j}\in \mathbb {K}^{n\times n}\) is a matrix with squared entries. Then its \(4\times 4\) minors are zero provided that the numbers \(a_{i,j}\) satisfy the Plücker relations defining the Grassmannian \({{\,\mathrm{Gr}\,}}_{2,n}\). An elementary direct proof was shown to us by Darij Grinberg (see [18]).
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Acknowledgements
The project whose results are presented here started with a research in pairs meeting at the Centro de Giorgi in Pisa in February 2018. We thank the Institute for a pleasant stay in a stimulating research environment. We also thank Aldo Conca, Raul Epure, Darij Grinberg, Delphine Pol and Karen Yeats for helpful comments. We are grateful to the referees for a careful reading of the manuscript and resulting improvements to the exposition.
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GD was supported by NSERC of Canada. MS was supported by the German Research Foundation (DFG) in the collaborative research center TRR 195 under Project B05 (#324841351). UW was supported by the NSF Grant DMS1401392 and by the Simons Foundation Collaboration Grant for Mathematicians #580839.
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Denham, G., Schulze, M. & Walther, U. Matroid connectivity and singularities of configuration hypersurfaces. Lett Math Phys 111, 11 (2021). https://doi.org/10.1007/s11005020013523
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Keywords
 Configuration
 Matroid
 Singularity
 Feynman
 Kirchhoff
 Symanzik
 Cohen–Macaulay
 Determinantal
Mathematics Subject Classification
 Primary 14N20
 Secondary 05C31
 14B05
 14M12
 81Q30