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Matroid connectivity and singularities of configuration hypersurfaces

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 non-smooth 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 3-connected, 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.


Feynman diagrams
A basic problem in high-energy 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 ∈ 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 T ∈T G e / ∈T x e of G where 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 2-forests 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 x e 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 well-behaved, 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 K. The same matroid can admit different realizations, which, in turn, give rise to different configuration poly-nomials (see Example 5.3). The matroid (basis) polynomial is a competing object, which is assigned to any, even non-realizable, 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 matrix-tree 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 ⊆ K E be a realization of a matroid M of rank rk M = dim W on a set E (see Definition 2.14). Fix coordinates x E = (x e ) e∈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 B M denotes the set of bases of M and the coefficients c W ,B ∈ K * depend of the realization W . The configuration hypersurface defined by ψ W is the scheme It can be seen as the first degeneracy scheme of Q W (see Definition 4.9). The second degeneracy scheme W ⊆ K E of Q W , defined by the submaximal minors of Q W , is a subscheme of the Jacobian scheme W ⊆ K E of X W , defined by ψ W and its partial derivatives (see Lemma 4.12). The latter defines the non-smooth locus of X W over K, which is the singular locus of X W if K is perfect (see Remark 4.10). Patterson showed W and 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 W is typically not reduced (see Example 5.1), whereas 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 M be a matroid on the set E with a linear realization W ⊆ K E over a field K. Then the configuration hypersurface X W is reduced and generically smooth over K. Moreover, the second degeneracy scheme W is also reduced and agrees with red W , the non-smooth locus of X W over K. Unless K has characteristic 2, the Jacobian scheme W is generically reduced.
Suppose now that M is connected. Then X W is integral unless M has rank zero. Suppose in addition that the rank of M is at least 2. Then W is Cohen-Macaulay of codimension 3 in K E . If, moreover, M is 3-connected, then W is integral.
Note that X W = ∅ if rk M = 0 and W = ∅ = W if rk M ≤ 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 M is disconnected even after deleting all loops, then W and hence W has codimension 2 in 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 2-separations (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 self-contained 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 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 W . The codimension of W in K E is therefore bounded by 3 and W is Cohen-Macaulay in case of equality (see Proposition 4.19). In this case, W is pure-dimensional, 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 non-disconnective 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 W \H associated with the deletion M\H has codimension 3 we prove the same for W (see Lemma 4.22). Applied to a non-disconnective H it follows with Patterson's result that 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 non-disconnective handle (see Corollary 4.26).
In case ch K = 2, generic reducedness of W implies (generic) reducedness of W . The schemes W and 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 W . Our proof proceeds by induction on the cardinality |E| of the underlying set E of the matroid M.
Unless M a circuit, the handle decomposition guarantees the existence of a nondisconnective handle H . In case H = {h} has size 1, the scheme W \h associated with the deletion M\h is the intersection of W with the divisor x e (see Lemma 4.29). This serves to recover generic reducedness of W from W \h (see Lemma 4.30). The same argument works if H does not arise from a handle decomposition.
This leads us to consider non-disconnective 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 matroid-theoretic ingredient, we show that the number of non-disconnective 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 non-disconnective handles of size 2 (see Lemmas 2.10 and 2.25). Here an explicit calculation shows that W is reduced in the torus (K * ) 6 (see Lemma 4.28). The corresponding result for W holds only if ch K = 2.
Suppose now that M is not a circuit and has no non-disconnective handles of size 1. Then M\e might be disconnected for all e ∈ E and does not qualify for an inductive step. In this case, we aim instead for contracting W by a suitable subset G E which keeps M connected. In the partial torus K F × (K * ) G where F := E\G, the scheme W /G associated with the contraction M/G relates to the normal cone of W along the coordinate subspace Lemma 4.31). To induce generic reducedness from W /G to 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 Rees I R with respect to an ideal I R, and the normal cone by the associated graded ring gr I R (see Definition 4.6). Passing through Rees I R, we recover generic reducedness of R along V (I ) from generic reducedness of gr I R (see Definition 4.3 and Lemma 4.7). By assumption on M, there are at least 3 more elements in E than maximal handles (see Proposition 2.12), and 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 G for which all generic points of W are along V (x F ) (see Lemma 4.34). This yields generic reducedness of W in this case (see Lemma 4.32). A slight modification of the approach also covers the generic points outside the torus (K * ) 6 if M is the prism matroid. The case where M is a circuit is reduced to that where M is a triangle by successively contracting an element of E (see Lemma 4.33). In this base case W is a reduced point, but W is reduced only if ch K = 2 (see Example 4.14).
Finally, suppose that M is a 3-connected matroid. Here we prove that W is irreducible and hence integral, which implies that is irreducible (see Theorem 4.37). We first observe that handles of (co)size at least 2 are 2-separations (see Lemma 2.4.(e)). It follows that the handle decomposition consists entirely of non-disconnective 1handles (see Proposition 2.5) and that all generic points of W lie in the torus (K * ) E (see Corollary 4.27). We show that the number of generic points is bounded by that of W \e for all e ∈ E (see Lemma 4.30). Duality switches deletion and contraction and identifies generic points of W and W ⊥ (see Corollary 4.18). Using Tutte's wheels-and-whirls theorem, the irreducibility of W can therefore be reduced to the cases where M is a wheel W n or a whirl W n for some n ≥ 3 (see Example 2.26 and Lemma 4.38). For fixed n, we show that the schemes X W , W and W are all isomorphic for all realizations W of W n and W n (see Proposition 4.40). An induction on n with an explicit study of the base cases n ≤ 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 Min P and Max P the set of minima and maxima of a poset P. Let M be a matroid on a set E =: E M . We use this font throughout to denote matroids. With 2 E partially ordered by inclusion, M can be defined by a monotone submodular rank function (see [26,Cor. 1.3.4 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]): • flats For instance (see [26,Lem. 1.3.3]), for any subset S ⊆ E, The closure operator of M is defined by (see [26,Lem. 1.4.2]) The following matroid plays a special role in the proof of our main result. The name comes from the observation that its independent sets I M are the affinely independent subsets of the vertices of the triangular prism (see Fig. 1).
The elements of E\ B M and B M are called loops and coloops in M, respectively. A matroid is free if E ∈ B M , that is, every e ∈ E is a coloop in M. By a k-circuit in M we mean a circuit C ∈ C M with |C| = k elements, 3-circuits are called triangles.
The circuits in M give rise to an equivalence relation on E by declaring e, f ∈ E equivalent if e = f or e, f ∈ C for some C ∈ C M (see [26,Prop. 4.1.2]). The corresponding equivalence classes are the connected components of M. If there is at most one such a component, then M is said to be connected. The connectivity function of M is defined by It is called exact if the latter is an equality. The connectivity λ(M) of M is the minimal k for which there is a k-separation of M, or λ(M) = ∞ if no such exists. The matroid M is said to be k-connected if λ(M) ≥ 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 M 1 ⊕ M 2 of matroids M 1 and M 2 is the matroid on E M 1 E M 2 with independent sets The sum is proper if E M 1 = ∅ = E 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 ⊆ E be any subset. Then the restriction matroid M| 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]) In §2.3, E will be a basis and E ∨ the corresponding dual basis. We often identify The complement of a subset S ⊆ E corresponds to The dual matroid M ⊥ is the matroid on E ∨ with bases (2.10) We use ν −1 in place of (2.8) for M ⊥ , so that S ⊥⊥ = S. For subsets F ⊆ E and G ⊆ E ∨ , one can identify (see [26, 3.1.1]) Various matroid data of M ⊥ is also considered as codata of M. A triad of M is a 3-cocircuit of M, that is, a triangle of M ⊥ .

Example 2.2 (Uniform matroids)
. The uniform matroid U r ,n of rank r ≥ 0 on a set E of size |E| = 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, U ⊥ r ,n = U n−r ,n for all 0 ≤ r ≤ n. Informally, we refer to a matroid M on E for which E ∈ C M , and hence, C M = {E}, as a circuit, and as a triangle if |E| = 3. It is easily seen that such a matroid is U n−1,n where n = |E|, and that λ(U n−1,n ) = 2.

Handle decomposition
In this subsection, we investigate handles as building blocks of connected matroids. which refines the partition of C M into connected components. In particular, each circuit is a disjoint union of maximal handles. For any subset F ⊆ E, (2.5) yields an inclusion

Definition 2.3 (Handles). Let M be a matroid. A subset
Whence X ∪ H is a 2-separation of M, a contradiction.
The following notion is the basis for our inductive approach to connected matroids.
By Lemma 2.4.(b) and (2.5), a handle decomposition yields circuits (2.12) Conversely, it can be constructed from a suitable sequence of circuits. A handle decomposition of length 2 is given by Note that all handles are proper, maximal, separating 2-handles. Proof There is a sequence of circuits C 1 , . . . , C k ∈ C M which defines a filtration . . , k (see [13]). The hypothesis Then there exists some d ∈ C i \(C ∪ F i−1 ). By the strong circuit elimination axiom (see [26,Prop. 1.4.12 In the sequel, we develop a bound for the number of non-disconnective handles.   (2.5)). In other words, M\H is connected. If H ∈ H M is a handle, then H is therefore non-disconnective.
Otherwise, there is a circuit C ∈ C M such that ∅ = C ∩ H = H . In particular, H ⊆ C since otherwise C ∩ H = ∅ and C ∈ C M\H (see (2.5)) which would contradict H ∈ H M\H . This means that C connects H with C ∩ H . We may therefore replace H by ∅ = H \C H and iterate. Then H ∈ H M after finitely many steps. Let C ∈ C M be any circuit with C = C = C. By incomparability of circuits, C C and hence H ⊆ C since H is a handle. By Lemma 2.4.(d), we may assume that |H | = 1. Then H ⊆ C (see [26,§1.1,Exc. 5]). In particular, H ∈ H M is a third non-disconnective handle. If H ∪ H ⊆ C is an equality, then also H ∈ H M is a non-disconnective handle and H H H is the handle decomposition. Otherwise, C witnesses the fact that H , H and ∅ = C ∩ H = H are in the same connected component of M| C (see (2.5)). If H \C ∈ H M is a handle, then it is therefore non-disconnective. Otherwise, iterating yields a third non-disconnective In particular, M is connected with a handle decomposition Here all 4 maximal handles are non-disconnective and the inequality in Lemma 2.10 is strict. This can happen because M is not a graphic matroid (see Lemma 2.25).
Proposition 2.12 (Lower bound for non-disconnective handles). Let M be a connected matroid with a handle decomposition of length k ≥ 2. Then M has at least k + 1 (disjoint) non-disconnective handles.
Proof We argue by induction on k. The base case k = 2 is covered by Lemma 2. 10. Suppose now that k ≥ 3. By hypothesis (see Definition 2.6), H k ∈ H M is a nondisconnective handle and the connected matroid M\H k = M| F k−1 has a handle decomposition of length k − 1. By induction, there are k (disjoint) non-disconnective handles H 0 , . . . , H k−1 ∈ H M\H k . Since k ≥ 3 and handles are non-empty, We conclude this section with an observation.
Proof Pick e ∈ E. Since M is connected, E is the union of all circuits e ∈ C ∈ C M . Suppose that there are only 2-circuits. Then E = cl M (e) (see [26,Prop. 1.4.11.(ii)]) and hence rk M = 1 (see (2.2)), a contradiction.

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 K and denote the K-dualizing functor by We consider a finite set E as a basis of the based K-vector space K E and denote by E ∨ = (e ∨ ) e∈E the dual basis of (2.13) By abuse of notation, we set S ∨ := (e ∨ ) e∈S for any subset S ⊆ E. We consider configurations as defined by Bloch, Esnault and Kreimer (see [6, §1]).
Let M be a matroid and W ⊆ K E a configuration (over K Since E ∨ | W generates W ∨ , we have (see (2.14))

Remark 2.15 (Matroids and linear algebra). The notions in matroid theory
. , e n } and set w i j := w i e j for j = 1, . . . , n. Then these vectors form the columns of the coefficient matrix A = (w i j ) i, j ∈ K r ×n of w. By construction, W is the row span of A. The matroid rank rk M (S) of any subset S ⊆ E now equals the K-linear rank of the submatrix of A with columns S (see (2.1) and (2.14)). An element e ∈ E is a loop in M if and only if column e of A is zero; e is a coloop in M if and only if column e is not in the span of the other columns.

Remark 2.16 (Classical configurations). Suppose that M W has no loops, that is,
Then the images of the e ∨ | W in PW ∨ form a projective point configuration in the classical sense (see [19]). Dually, the hyperplanes ker(e ∨ ) ∩ 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 ⊆ E gives rise to an inclusion and a projection of based K-vector spaces.

Definition 2.17 (Realizations of matroid operations).
Let W ⊆ K E be a realization of a matroid M, and let F ⊆ E be any subset.
Proof By definition (see Definition 2.17.(a) and (c)), By the direct sum hypothesis, W i and W | E i realize the same matroid (see (2.3), (2.4) and (2.6)) (2.15)) and the claim follows.
Example 2.20 (Realizations of uniform matroids). Let W ⊆ K E be the row span of a matrix A ∈ K r ×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 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.
is a pair of finite sets V of vertices and E of (unoriented) edges where each edge e ∈ 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 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 (non-unique) connected graph (see [26,Prop. 1.2.9]). Unless specified otherwise, we therefore assume that G is connected. Then the bases of 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
is a subset of V whose removal (together with all incident edges) disconnects G. If G has at least one pair of distinct non-adjacent vertices, then  Graphic matroids have realizations derived from the edge-vertex incidence matrix of the graph (see [6, §2]). A choice of orientation on the edge set E turns the graph G into a CW-complex. This gives rise to an exact sequence where H • := H • (G, K) denotes the graph homology of G over K. The dual exact sequence Definition 2.23 (Graph configurations). We call the image of δ ∨ the graph configuration of the graph G over K. Note that it is independent of the orientation chosen to define δ in (2.18).
For any S ⊆ E, the sequence (2.18) induces a short exact sequence 0 By definition of M G and 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 ch K = 0 (see [26,Lem. 5.1.4]) which makes M G a regular matroid. By construction, Its rows generate the graph configuration W G realizing the prism matroid (see Example 2.22).
identifies with the circuits of the prism matroid. It follows that M must be the prism matroid.
Let W ⊆ K E be any realization of M. Then dim W = rk M = 4 (see (2.15) and (2.17)). Pick a basis w = (w 1 , . . . , 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 = w 2 3 and w 2 Since C 3 is a circuit, suitably replacing w 3 , w 4 ∈ w 3 , w 4 , reordering H 3 and scaling e 1 , e 3 makes where w 1 1 , w 2 3 , w 3 5 = 0. Now suitably scaling first w 1 , w 2 , w 3 and then e 2 , e 4 , e 6 makes The following classes of matroids play a distinguished role in connection with 3-connectedness.
Example 2.26 (Wheels and whirls). For n ≥ 2, the wheel graph W n in Fig. 3 is obtained from an n-cycle, 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 Z n := Z/nZ = {1, . . . , n}.
For n ≥ 3, the wheel matroid is the graphic matroid W n := M W n on E. For n ≥ 2, the whirl matroid is the (non-graphic) matroid on E obtained from M W n by relaxation of the rim R, that is, In terms of circuits, this means that The matroids W n and W n are 3-connected (see [26,Exa. 8.4.3]) of rank rk W n = n = rk W n .
For each i ∈ Z n , {s i , r i , s i+1 } is a triangle and {r i , r i+1 , s i+1 } a triad. Conversely, this property enforces M ∈ W n , W n for any connected matroid M on E (see [29, (6 We describe all realizations of wheels and whirls up to equivalence. In particular, we recover the well-known fact that whirls are not binary.
Proof Since S ∈ B M , we may assume that the coefficients of s j in w i form an identity matrix, that is, . . , s n successively yields (2.20). The claim on t follows from R ∈ C W n and R ∈ B W n , respectively.

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 ⊆ K E be a configuration, and let S ⊆ E be any subset. Compose the associated inclusion map with π S to a map (see (2.16)) Fix an isomorphism and set c 0 := id 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 ∈ K * .
Up to sign or ordering E, we identify as based vector spaces. Suppose that |S| = dim W . Then the determinant is defined up to sign. Its square is defined up to a factor c 2 for some c ∈ K * independent of S. Note that det α 0,∅ = id K and hence c 0,∅ = 1. By definition (see (2.14)), Consider the dual basis E ∨ = (e ∨ ) e∈E of E as coordinates on K E , Given an enumeration of E = {e 1 , . . . , e n }, we write For any subset S ⊆ E, we set

Remark 3.3 (Well-definedness of configuration polynomials)
. Any two isomorphisms c W (see (3.2)) differ by a nonzero multiple c ∈ K * . Using the isomorphism c · c W in place of c W replaces ψ W by c 2 · ψ W . In other words, ψ W is well-defined up to a nonzero constant square factor. Whenever ψ W occurs in a formula, we mean that the formula holds true for a suitable choice of such a factor.
Realizations of U 2,n are treated in Example 5.4.
In the following, we put matroid connectivity in correspondence with irreducibility of configuration polynomials.
Proof First suppose that M = M 1 ⊕M 2 is disconnected with underlying proper partition (2.3)). It follows that ψ W = ψ W 1 · ψ W 2 . This factorization is proper if M and hence each M i has no loops (see Remark 3.5). Thus, ψ W is reducible in this case.
Suppose now that ψ W is reducible. Then with ψ i homogeneous non-constant for i = 1, 2. Since ψ W is a linear combination of square-free monomials (see Definition 3.2), this yields a proper partition (2.4) and (3.9)). As above, (3.6)). It follows that B M ⊇ B M 1 ⊕M 2 as well.
So M = M 1 ⊕ M 2 is a proper decomposition and M is disconnected. This proves the equivalence and the particular claims follow.
We use the following well-known fact from linear algebra. Abbreviate Tensored with

respectively, it induces identifications
Consider a commutative diagram of finite dimensional 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]).
where the middle isomorphism is induced by (2.8). This yields a commutative diagram (Remark 3.9 and (2.15)) Using The coefficients of the configuration polynomial satisfy the following restrictioncontraction formula.
Taking exterior powers yields (see Remark 3.9 and (2.15)) 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 ∂ e ψ W was proven by Patterson (see [27,Lem. 4.4]).

Proposition 3.12 (Deletion-contraction for configuration polynomials). Let W ⊆ K E be a realization of a matroid M, and let e ∈ E. Then
where ψ W | e = x e if e is not a loop in M. In particular, for some c ∈ K * . If e is a (co)loop, then W /e = W \e (see Remark 2.18.(a)). The claimed formulas follow.
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.   The following result describes the behavior of configuration polynomials when passing to an elementary quotient.

any lift of ϕ with a sign ± determined by a Laplace expansion.
Proof Set V := W ⊥ and V ϕ := W ⊥ ϕ and consider the commutative diagram with short exact rows and columns Dualizing and identifying the two copies of 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 Due to (3.19) the maps α V ϕ ,S (see (3.1)) and agree after applying rk M ⊥ +1 . Laplace expansion thus yields

Graph polynomials
We continue the discussion of graphic matroids from §2.4 and consider their configuration polynomials.
Replacing x T by x E\T defines the (first) Symanzik polynomial ψ ⊥ G of a graph G over K. We refer to ψ G and ψ ⊥ G as (first) graph polynomials.
By (2.17), we have ψ G = ψ W for any totally unimodular realization W of 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).
Let G = (E, V ) be a graph. A 2-forest in G is an acyclic subgraph T of G with |V | − 2 edges. Any such T = {T 1 , T 2 } has 2 connected components T 1 and T 2 . We denote by T 2 G the set of all 2-forests in G.

Definition 3.18 (Second graph polynomials).
The second Kirchhoff polynomial of a graph G over K is the polynomial depending on a momentum 0 = p ∈ ker σ for G over K (see (2.18)). Note that Replacing the 2-forests T 1 T 2 by cut sets E\(T 1 T 2 ) defines the second Symanzik polynomial ψ ⊥ G ( p) of a graph G over K (see [27,Def. 3.6]). We refer to ψ G ( p) and ψ ⊥ G ( 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).

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 non-smooth locus of the hypersurface defined by the corresponding configuration polynomial.

Definition 3.20 (Configuration forms). Let μ K denote the multiplication map of K.
Consider the generic diagonal bilinear form on K E , Let W ⊆ K E be a configuration. Then the configuration (bilinear) form of W is the restriction of Q K E to W , Alternatively, it can be seen as the composition of canonical maps Its image is the kth Fitting ideal Fitt k coker Q W (see [16, §20.2]) and defines the k −1st degeneracy scheme of Q W . We set Note the different fonts used for M W and M W (see Definition 2.14).

Remark 3.21 (Configuration forms as matrices). With respect to a basis
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 = U w for some U ∈ Aut K W . Then We often consider Q W as a matrix Q w determined up to conjugation.
Lemma 3.23 recovers the polynomial det Q W = ψ W in Example 3.17.
The following result describes the behavior of Fitting ideals of configuration forms under duality. We consider the torus (3.21)

Proposition 3.25 (Duality and cokernels of configuration forms). Let W ⊆ K E be a configuration. Then there is an isomorphism over ζ E ,
where the indices denote localization (see (3.8)). In particular, this induces an isomorphism Proof Consider the short exact sequence We identify K E = K E ∨∨ and K E /W = W ⊥∨ , and we abbreviate Exactness of the columns is due to det Q W = ψ W = 0 (see Lemma 3.23 and Remark 3.5). Composing the middle vertical isomorphism over ζ E with (taking preimages along) the dashed compositions yields the claimed isomorphism by a diagram chase.
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.  • If e is a loop, then w i e = 0 for all i = 1, . . . , r and hence W \e = W = W /e. • If e is not a loop, then we may adjust w 1 , . . . , w r such that w i e = δ i,r for all i = 1, . . . , r and then w 1 , . . . , w r −1 is a general basis of W /e.
• If e is a coloop, then we may adjust w r = e and π E\{e} identifies w 1 , . . . , w r −1 with a basis of W \e = W /e.
In the latter case, (3.24) and the claimed equalities follow (see Lemma 3.23).
It remains to consider the case in which e is not a (co)loop. Then ι E\{e} and π E\{e} (see (2.16)) identify w 1 , . . . , w r −1 and w 1 , . . . , w r with bases of W /e and W \e, respectively. Hence, (3.25) where both the entry a and column b are independent of x e . We consider two cases.
This proves the claimed equalities also in this case (see Lemma 3.23) and the particular claim follows.
As an application of Lemma 3.23, we describe the behavior of configuration polynomials under 2-separations.

Proposition 3.27 (Configuration polynomials and 2-separations). Let W ⊆ K E be a realization of a connected matroid M. Suppose that E
Proof We adopt the notation from [30, §8.2]. Extend a basis B 2 ∈ B M| E 2 to a basis B ∈ B 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\B 2 , B 2 , E 1 \B, E 2 \ B 2 , and rk D = 1. After suitably ordering and scaling B 2 , E 1 \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 ∈ E the index of the column (0 1 b) which involves Laplace expansion of ψ W = det Q W (see Lemma 3.23) along the eth column yields the claimed formula.

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 ch K = 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 non-experts 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 Min Spec R and Max Spec R the sets of minimal and maximal elements of the set Spec R of prime ideals of R with respect to inclusion. The subset Ass R ⊆ Spec R of associated primes of R is finite and Min Spec R ⊆ Ass R (see [24,Thm. 6.5]). One says that R is catenary if every saturated chain of prime ideals joining p, q ∈ Spec R with p ⊆ q has (maximal) length height(q/p) (see [24,31]). We say that R is equidimensional if it is catenary and If R is a finitely generated K-algebra, then these two conditions reduce to (see [7,Thm. 2.1.12] and [24,Thm. 5.6]) which implies in particular that Ass R = Min Spec R. It follows that pure-dimensional finitely generated K-algebras are equidimensional.
The following lemma applies to any equidimensional finitely generated K-algebra.

Lemma 4.1 (Height bound for adding elements). Let R be a Noetherian ring such that R m is equidimensional for all
(a) All saturated chains of primes in p ∈ Spec R have length height p.
(b) For any p ∈ Spec R, x ∈ R and q ∈ Spec R minimal over p + x , Proof (a) Take two such chains of length n and n starting at minimal primes p 0 and p 0 , respectively. Extend both by a saturated chain of primes of length m containing p and ending in a maximal ideal m. Since R 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, height(q/p) ≤ 1. Take a chain of primes in p of length height p and extend it by q if p = q. By (a), this extended chain has length height q and the claim follows.

Lemma 4.2 (Equidimensional finitely generated algebras and localization). Let R be an equidimensional finitely generated K-algebra and x
Proof Any minimal prime ideal of R x is of the form p x where p ∈ Min Spec R with x / ∈ p. By the Hilbert Nullstellensatz (see [24,Thm. 5.5]), This yields an m ∈ Max Spec R such that p ⊆ m x and hence p x ⊆ m x ∈ Max Spec R x . Since R and hence R x is a finitely generated K-algebra, by equidimensionality of R. The claim follows.

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 p is reduced for all p ∈ Min 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 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 = 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 pure-dimensional.
Proof Since R is pure-dimensional, Ass R = Min Spec R, and hence, R becomes a subring of localizations (see [24,Thm. 6 The latter ring is reduced since R is generically reduced, and the claim follows.

Lemma 4.5 (Reducedness and reduction). Let (R, m) be a local Noetherian ring. Suppose that R/t R is reduced for a system of parameters t. Then R is regular and, in particular, an integral domain and reduced.
Proof By hypothesis, R/t R is local Artinian with maximal ideal m/t R. Reducedness makes R/t R a field, and hence, m = t R. By definition, this means that R is regular. In particular, R is an integral domain and reduced (see [24,Thm. 14.3]).
The associated graded algebra is the R/I -algebra Lemma 4.7 (Generic reducedness from associated graded ring). Let R be a Noetherian d-dimensional ring, I R an ideal, S := Rees I R andR := gr I R.
(a) Suppose R is an equidimensional finitely generated K-algebra. Then S is a (d +1)equidimensional finitely generated K-algebra. (b) If S is (d + 1)-equidimensional and I = R, thenR is d-equidimensional. (c) If S is equidimensional andR 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 ⊆ R[t ±1 ]. By definition, formation of the Rees ring commutes with base change. After base change to R/p for some p ∈ Min Spec R, we may assume that R is a d-dimensional integral domain. Then S is a (d + 1)-dimensional integral domain (see [20,Thm. 5.1.4]). Since S is a finitely generated K-algebra (as R is one), S is equidimensional.
To check injectivity, consider ]. Then 0 = x y i ∈ R for all i and y j ∈ R\p for some j. It follows that 0 = x/1 ∈ R p . Combining (4.1) and (4.2) reducedness of R p follows from reducedness of Sp. Suppose now that V (p) ∩ V (I ) = ∅ and hence (the subscript denoting graded parts) implies thatp + t S = S. Let q ∈ Spec S be a minimal prime ideal overp + t S.
No minimal prime ideal of S contains the S-sequence t ∈ q. By Lemma 4.1.(b), height q = 1 and q is minimal over t. This makes t a parameter of the localization S q . Under S/t S ∼ =R, the minimal prime ideal q/t S ∈ Spec(S/t S) corresponds to a minimal prime idealq ∈ SpecR. Suppose thatR is generically reduced. Then S q /t S q = (S/t S) q/t S ∼ =Rq is reduced. By Lemma 4.5, S q and hence its localization (S q )p q = Sp is reduced. Then also R p is reduced, as shown before.

Graded Cohen-Macaulay rings
Let (R, 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 m. For any p ∈ Spec R, denote by p * ∈ Spec R the maximal graded ideal contained in p (see [7, Lem. 1.5.6.(a)]). For any p ∈ Spec R, there is a chain of maximal length of graded prime ideals strictly contained in p (see [7,Lem. 1.5.8]). If m / ∈ Max Spec R, then such a chain for n ∈ Max Spec R ends with m n. It follows that If I R is a graded ideal and p ∈ Spec R minimal over I , then p/I ∈ Min Spec(R/I ) ⊆ Ass(R/I ), and hence, p is graded.
The following lemma shows in particular that * local Cohen-Macaulay rings are pure-and equidimensional. height I m = codim I m (4.7) Using (4.3), (4.4) for I = p and bijection (4.5), it follows that R is pure-dimensional: 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/p, I is equidimensional if and only if dim(R/I ) = dim(R/p) for all prime ideals p ∈ Spec R minimal over I . The particular claim then follows by (4.6) for I and p.

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 Spec R, whose underlying set consists of all prime ideals of R. We refer to elements of Min Spec R as generic points, of Ass R as associated points, and of Ass R\ Min Spec R as embedded points of Spec R. An ideal I R defines a subscheme Spec(R/I ) ⊆ Spec R.
By abuse of notation we identify

Definition 4.9 [Configuration schemes]
Let W ⊆ 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 ψ W . We call the subschemes (see Definition 3.20) the Jacobian scheme of X W and the second degeneracy scheme of Q W .     If ch K = 2, the same holds for W due to the Euler identity (see Remark 4.10). Otherwise, the non-redundant quadratic generator ψ W of J W can make W nonreduced (see Example 4.14). With respect to this basis, we compute

Lemma 4.12 (Inclusions of schemes
It follows that W is a reduced point.
On the other hand, The matrix expressing the linear generators and W is a non-reduced point.   Lemma 2.19). Then X W is the reduced union of integral schemes X W i × K E\E i , and W is the union of W i × K E\E i and integral schemes X W i × X W j × K E\(E i ∪E j ) for i = j. The same holds for replaced by . In particular, X W is generically smooth over K.
Proof Proposition 3.8 yields the claim on X W (see Remark 3.5). For the claims on W and W , we may assume that n = 2 with M 1 possibly disconnected. The general case then follows by induction on n.
By Proposition 3.8 and Definition 3.20, ψ W = ψ W 1 · ψ W 2 and Q W = Q W 1 ⊕ Q W 2 . Then Lemma 4.15 yields and hence, The same holds for J and replaced by M and , respectively.
Suppose now that M is connected. By Proposition 3.12, ψ W ∂ e ψ W for any e ∈ E and hence W X W . The particular claim follows.
Patterson proved the following result (see [27,Thm. 4.1]). While Patterson assumes ch K = 0 and excludes the generator ψ W ∈ J W , his proof works in general (see Remark 4.10). We give an alternative proof using Dodgson identities. In particular, W and W have the same generic points, that is, Proof Order E = {e 1 , . . . , e n } and pick a basis w = (w 1 , . . . , w r ) of W . We may assume that its coefficients with respect to e 1 , . . . , e r form an identity matrix, that is, In particular, W , W , W ⊥ and W ⊥ have the same generic points in T E ∼ = T E ∨ .
Proof Propositions 3.10 and 3.25 yield the statements for X W and W . The statement for W follows using that ζ E (see (3.21)) identifies x e ∂ e = −x e ∨ ∂ e ∨ for e ∈ E. The particular claim follows with Theorem 4.17.

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 non-disconnective handle.
Proof By Remark 3.4 and Corollary 3.13, we may assume that has the form (3.14).
(a) Using that ψ W is a linear combination of square-free monomials (see Definition 3.2), (c) This follows from By symmetry, it follows that x 2 x 4 x 6 · p ⊆ M W and hence Using ψ W from Example 3.17, one computes that By symmetry, it follows that 2 · x 2 2 x 2 4 x 2 6 · p ⊆ J W and hence More details on the prism matroid can be found in Example 5.1. By the dimension hypothesis, Lemma 4.8 and (4.13), it follows thatq is minimal over both J W \e andJ W . The former means thatq ∈ Min W \e . The set γ (p) of all suchq is non-empty and satisfies condition (4.11). Denote by t ∈ K[ W ] the image of x e . Then q / ∈ Min K[ W ] by hypothesis and q is minimal over t sinceq is minimal overJ W . This makes t is a parameter of the localization

Lemma 4.29 (Reduction and deletion of non-(co)loops
The inclusion (4.13) gives rise to a surjection of local rings (4.14) Suppose now that W \e is generically reduced. Then K[ W \e ]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 p q . Thus, K[ W ] p = R p q is reduced and p is uniquely determined byq. This uniqueness is condition (4.12). The particular claim follows immediately.
The preceding arguments remain valid if and J are replaced by and M, respectively: Lemma 4.29 applies in both cases. Lemma 4.31 (Initial forms and contraction of non-(co)loops). Let W ⊆ K E be a realization of a matroid M. Suppose E = F G is partitioned in such a way that M/G is obtained from M by successively contracting non-(co)loops. For any ideal , denote by J inf the ideal generated by the lowest x F -degree parts of the elements of J .
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 ∈ G. This yields a basis of W extending a basis w 1 , . . . , w s of W /G such that   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 = 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 π −1 induces Suppose now that W /G and hence W /G × T G is generically reduced. For any p ∈ Min SpecR, this makes K[ W /G × T G ] p a field and due to alsoR p is a field. It follows thatR is generically reduced. By Lemma 4.7, R is then generically reduced along V (I ). This means that The preceding arguments remain valid if and J are replaced by and M, respectively: Lemma 4.31 applies in both cases. Proof We proceed by induction on |E|. The case |E| = 3 is covered by Example 4.14; here we use ch K = 2.
Suppose now that |E| > 3. Let p ∈ Min W be a generic point of W . By Lemma 4.24, p = x e , x f , x g for some e, f , g ∈ H with e = f = g = e. Pick d ∈ E\{e, f , g}. Then E\{d} ∈ C M/d and hence W /d is generically reduced by induction. By Lemmas 4.2 and 4.32 Thus, W is reduced at p. It follows that W is generically reduced.
By Theorem 4.17, W has the same generic points as W . Therefore, the preceding arguments remain valid if is replaced by . (4.16) If > 2 and p ∈ V (x e ), then (4.16) holds with 3 replaced by 2. In either case pick Add to F all f ∈ E with x f ∈ q. This does not affect (4.17). Then x g / ∈ q and hence x g / ∈ p for all g ∈ G := E\F = ∅. In other words, . Suppose now that p ∈ W and hence rk(M/G) ≥ 2. Then W /G is generically reduced by hypothesis, and p ∈ W ∩ D(x G ) specializes to a point in V (x F ) ∩ D(x G ) by (4.18). By Theorem 4.25 and Lemma 4.2, W , W ∩ D(x G ) and W /G are equidimensional of codimension 3. By Lemma 4.8, height p = 3 means that p ∈ Min W . By Lemma 4.32, W is thus reduced at p. The claims follow.
The preceding arguments remain valid if is replaced by .

Integrality of degeneracy schemes
In this subsection, we prove the following companion result to Proposition 3.8 as outlined in §1.4. (4.20) It follows that integrality is equivalent for W and W ⊥ . In particular, we may also assume that rk M ⊥ ≥ 3. We proceed by induction on |E|. Suppose that M is not a wheel or a whirl. Since rk M ≥ 3, Tutte's wheels-and-whirls theorem (see [26,Thm. 8.8.4]) yields an e ∈ E such that M\e or M/e is again 3-connected. In the latter case, we replace W by W ⊥ and use (2.11). We may thus assume that M\e is 3-connected. Then W \e is integral by induction hypothesis. Note that Min W ⊆ D(x e ) by (4.20). By Theorem 4.25, W and W \e are equidimensional of codimension 3. By Remark 4.13.(a) and Lemma 4.30, W = ∅ and |Min W | ≤ Min W \e = 1. It follows that W is integral. Lemma 4.39 (Turning wheels). Let W ⊆ K E be the realization of W n from Lemma 2.27. Then the cyclic group Z n acts on X W , W and W by "turning the wheel," induced by the generator 1 ∈ Z n mapping s i → s i+1 , r i → r i+1 , w i → w i+1 . (4.21) Proof By Lemma 2.27, W has a basis w = (w 1 , . . . , w n ) where w i = s i + r i − r i−1 for all i ∈ Z n . The assignment (4.21) stabilizes W ⊆ K E . The resulting Z n -action stabilizes ψ W and Q W , and hence J W and M W . As a consequence, it induces an action on X W , W and W .
The graph hypersurface of the n-wheel was described by Bloch, Esnault and Kreimer (see [6, (11.5)]). We show that it is also the unique configuration hypersurface of the n-whirl.
Proof Suppose that n is not minimal for M ∈ W n , W n to be defined. Let W be any realization of M/r n . Then W \s n is a realization of Let W be any realization of M and use the coordinates from (4.22). By Lemma 4.42.(b) and Corollary 4.26, W \s n has at most one generic point q in V (y n−1 , y n ) while all the others lie in T E\{s n } . By Corollary 4.18, the Cremona isomorphism identifies the latter with generic points of (W \s n ) ⊥ in T E ∨ \{s ∨ n } . Use (2.11) and Lemma 4.42.(a) to identify (M\s n ) ⊥ = M ⊥ /s ∨ n = M/r n , E ∨ \ s ∨ n = E\{r n }, and consider (W \s n ) ⊥ as a realization W of M/r n . By the above, W is integral with generic point in T E\{r n } . Thus, W \s n has a unique generic point q in T E\{s n } . To summarize, Min W \s n = q, q , q ∈ T E\{s n } , q ∈ V (y n−1 , y n ). By (4.11) in Lemma 4.30, we may assume that p = q and p = q whereĪ := (I + z n )/ z n .
Consider first the case where M = W n with n ≥ 4. By Remark 3.22, we may assume that W is the realization from Lemma 2.27. By Lemma 4.39, the cyclic group Z n acts on p, p by "turning the wheel." If it acts identically, then √ p + z i ⊇ y i−1 , y i for all i = 1, . . . , n and hence p + z 1 , . . . , z n = z 1 , . . . , z n , y 1 , . . . , y n .
This leads to a contradiction as before. Consider now the case where M = W n with n ≥ 5. For i = 1, . . . , n, denote by q i and q i the generic points of W \s i as in (4.23). By the pigeonhole principle, one of p and p , say p, is assigned to q i for 3 spokes s i . In particular, p is assigned to q i and q j for two non-adjacent spokes s i and s j . Then p + z i , z j ⊇ z i , z j , y i−1 , y i , y j−1 , y j .
This leads to a contradiction as before. The claim follows.

Examples
In this section, we illustrate our results with examples of prism, whirl and uniform matroids. in T 6 . By Corollary 4.26, there can be at most 3 more generic points symmetric to Over K = F 2 , their presence is confirmed by a computation in Singular (see [14]). It reveals a total of 7 embedded points in W . There is x 1 , . . . , x 6 , and 3 symmetric to each of x 3 , x 4 , x 5 , x 6 and x 1 , x 2 , x 3 + x 4 , x 5 + x 6 .
Moreover, W is not reduced at any generic point. Since the above associated primes are geometrically prime, the conclusions remain valid over any field K with ch K = 2.
A Singular computation over Q shows that W has exactly the above associated points for any field K with ch K = 0 or ch K 0. We expect that this holds in fact for ch K = 2.