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 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 \({{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\) 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
$$\begin{aligned} \forall \mathfrak {p}\in {{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R:\forall \mathfrak {m}\in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R:\mathfrak {p}\subseteq \mathfrak {m}\implies {{\,\mathrm{height}\,}}(\mathfrak {m}/\mathfrak {p})=\dim R. \end{aligned}$$
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])
$$\begin{aligned} \forall \mathfrak {p}\in {{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R:\dim (R/\mathfrak {p})=\dim R. \end{aligned}$$
We say that R is pure-dimensional if
$$\begin{aligned} \forall \mathfrak {p}\in {{\,\mathrm{Ass}\,}}R:\dim (R/\mathfrak {p})=\dim R, \end{aligned}$$
which implies in particular that \({{\,\mathrm{Ass}\,}}R={{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\). It follows that pure-dimensional 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]),
$$\begin{aligned} \bigcap {{\,\mathrm{Max}\,}}V(\mathfrak {p})=\mathfrak {p}. \end{aligned}$$
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,
$$\begin{aligned} \dim (R_x/\mathfrak {p}_x)={{\,\mathrm{height}\,}}(\mathfrak {m}_x/\mathfrak {p}_x)={{\,\mathrm{height}\,}}(\mathfrak {m}/\mathfrak {p})=\dim R \end{aligned}$$
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 pure-dimensional.
Proof
Since R is pure-dimensional, \({{\,\mathrm{Ass}\,}}R={{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R\), and hence, R becomes a subring of localizations (see [24, Thm. 6.1.(i)])
$$\begin{aligned} R\hookrightarrow \bigoplus _{\mathfrak {p}\in {{\,\mathrm{Ass}\,}}R}R_\mathfrak {p}=\bigoplus _{{{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}R}R_\mathfrak {p}. \end{aligned}$$
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])
$$\begin{aligned} {{\,\mathrm{Rees}\,}}_IR:=R[t,It^{-1}]\subseteq R[t^{\pm 1}]. \end{aligned}$$
The associated graded algebra is the R/I-algebra
$$\begin{aligned} {{\,\mathrm{gr}\,}}_IR:=\bigoplus _{i=0}^\infty I^i/I^{i+1}. \end{aligned}$$
Lemma 4.7
(Generic reducedness from associated graded ring). Let R be a Noetherian d-dimensional 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 d-equidimensional.
-
(c)
If S is equidimensional and \(\bar{R}\) is generically reduced, then R is generically reduced along V(I).
Proof
There are ring homomorphisms
$$\begin{aligned} R\rightarrow R[t]\rightarrow S\rightarrow S/tS\cong \bar{R}. \end{aligned}$$
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 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 \(\mathbb {K}\)-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 S-sequence. Since S is \((d+1)\)-equidimensional, \(\bar{R}\) is d-equidimensional 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 S-sequence \(t\in \mathfrak {q}\). 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
$$\begin{aligned} \dim R= {\left\{ \begin{array}{ll} \dim R_\mathfrak {m}&{} \text {if }\mathfrak {m}\in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R,\\ \dim R_\mathfrak {m}+1 &{} \text {if }\mathfrak {m}\not \in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R. \end{array}\right. } \end{aligned}$$
(4.3)
For any proper graded ideal \(I\lhd R\) also \((R/I,\mathfrak {m}/I)\) is \(^*\)local and
$$\begin{aligned} \mathfrak {m}\in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R\iff \mathfrak {m}/I\in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}(R/I). \end{aligned}$$
(4.4)
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])
$$\begin{aligned} {{\,\mathrm{Ass}\,}}R\rightarrow {{\,\mathrm{Ass}\,}}R_\mathfrak {m},\quad \mathfrak {p}\mapsto \mathfrak {p}_\mathfrak {m}. \end{aligned}$$
(4.5)
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 pure-dimensional and
$$\begin{aligned} {{\,\mathrm{height}\,}}I={{\,\mathrm{codim}\,}}I. \end{aligned}$$
(4.6)
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 pure-dimensional (see [7, Prop. 1.2.13]) and (see [7, Cor. 2.1.4])
$$\begin{aligned} {{\,\mathrm{height}\,}}I_\mathfrak {m}={{\,\mathrm{codim}\,}}I_\mathfrak {m}\end{aligned}$$
(4.7)
Using (4.3), (4.4) for \(I=\mathfrak {p}\) and bijection (4.5), it follows that R is pure-dimensional:
$$\begin{aligned} \forall \mathfrak {p}\in {{\,\mathrm{Ass}\,}}R:\dim R&= {\left\{ \begin{array}{ll} \dim R_\mathfrak {m}&{} \text {if }\mathfrak {m}\in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R,\\ \dim R_\mathfrak {m}+1 &{} \text {if }\mathfrak {m}\not \in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R,\\ \end{array}\right. }\\&= {\left\{ \begin{array}{ll} \dim (R_\mathfrak {m}/\mathfrak {p}_\mathfrak {m}) &{} \text {if }\mathfrak {m}\in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R,\\ \dim (R_\mathfrak {m}/\mathfrak {p}_\mathfrak {m})+1 &{} \text {if }\mathfrak {m}\not \in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R,\\ \end{array}\right. }\\&= {\left\{ \begin{array}{ll} \dim (R/\mathfrak {p})_{\mathfrak {m}/\mathfrak {p}} &{} \text {if }\mathfrak {m}\in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R,\\ \dim (R/\mathfrak {p})_{\mathfrak {m}/\mathfrak {p}}+1 &{} \text {if }\mathfrak {m}\not \in {{\,\mathrm{Max}\,}}{{\,\mathrm{Spec}\,}}R,\\ \end{array}\right. }\\&=\dim (R/\mathfrak {p}). \end{aligned}$$
Using (4.3) and (4.4), (4.6) follows from (4.7):
$$\begin{aligned} {{\,\mathrm{height}\,}}I&={{\,\mathrm{height}\,}}I_\mathfrak {m}={{\,\mathrm{codim}\,}}I_\mathfrak {m}\\&=\dim R_\mathfrak {m}-\dim (R_\mathfrak {m}/I_\mathfrak {m})\\&=\dim R_\mathfrak {m}-\dim (R/I)_{\mathfrak {m}/I}\\&=\dim R-\dim (R/I) ={{\,\mathrm{codim}\,}}I. \end{aligned}$$
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
$$\begin{aligned} \mathbb {K}^E={{\,\mathrm{Spec}\,}}\mathbb {K}[x]. \end{aligned}$$
Due to Lemma 4.8,
$$\begin{aligned} {{\,\mathrm{codim}\,}}_{\mathbb {K}^E}{{\,\mathrm{Spec}\,}}(\mathbb {K}[x]/I)={{\,\mathrm{height}\,}}I \end{aligned}$$
for any graded ideal \(I\unlhd \mathbb {K}[x]\).
Definition 4.9
Let \(W\subseteq \mathbb {K}^E\) be a configuration. Then the subscheme
$$\begin{aligned} X_W:={{\,\mathrm{Spec}\,}}(\mathbb {K}[x]/{\left\langle \psi _W\right\rangle })\subseteq \mathbb {K}^E \end{aligned}$$
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
$$\begin{aligned} J_W:={\left\langle \psi _W\right\rangle }+{\left\langle \partial _e\psi _W\;\big |\;e\in E\right\rangle }\unlhd \mathbb {K}[x] \end{aligned}$$
is the Jacobian ideal of \(\psi _W\). We call the subschemes (see Definition 3.20)
$$\begin{aligned} \Sigma _W:={{\,\mathrm{Spec}\,}}(\mathbb {K}[x]/J_W)\subseteq \mathbb {K}^E,\quad \Delta _W:={{\,\mathrm{Spec}\,}}(\mathbb {K}[x]/M_W)\subseteq \mathbb {K}^E, \end{aligned}$$
the Jacobian scheme of \(X_W\) and the second degeneracy scheme of \(Q_W\).
Remark 4.10
(Degeneracy and non-smooth 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 non-smooth 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)
$$\begin{aligned} X_W=X_{W{\setminus } e}\times \mathbb {A}^1,\quad \Sigma _W=\Sigma _{W{\setminus } e}\times \mathbb {A}^1,\quad \Delta _W=\Delta _{W{\setminus } e}\times \mathbb {A}^1. \end{aligned}$$
\(\square \)
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 non-redundant quadratic generator \(\psi _W\) of \(J_W\) can make \(\Sigma _W\) non-reduced (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
$$\begin{aligned} w^1:=e_1+e_3,\quad w^2:=e_2+e_3. \end{aligned}$$
With respect to this basis, we compute
$$\begin{aligned} Q_W&= \begin{pmatrix} x_1+x_3 &{} \quad x_3\\ x_3 &{} \quad x_2+x_3 \end{pmatrix},\\ M_W&={\left\langle x_1+x_3,x_2+x_3,x_3\right\rangle }={\left\langle x_1,x_2,x_3\right\rangle }. \end{aligned}$$
It follows that \(\Delta _W\) is a reduced point.
On the other hand,
$$\begin{aligned} \psi _W&=\det Q_W=x_1x_2+x_1x_3+x_2x_3,\\ J_W&={\left\langle \psi _W,x_1+x_2,x_1+x_3,x_2+x_3\right\rangle }. \end{aligned}$$
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,
$$\begin{aligned} J_W={\left\langle \psi _W,x_1-x_3,x_2-x_3\right\rangle }={\left\langle x_1-x_3,x_2-x_3,x_3^2\right\rangle } \end{aligned}$$
and \(\Sigma _W\) is a non-reduced 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
$$\begin{aligned} f\cdot J[x]+I[y]\cdot g={\left\langle f,g\right\rangle }\cap I[y]\cap J[x]\unlhd \mathbb {K}[x,y]. \end{aligned}$$
Proof
For the non-obvious 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
$$\begin{aligned} J_W&=\psi _{W_1}\cdot J_{W_2}[x_{E_1}]+J_{W_1}[x_{E_2}]\cdot \psi _{W_2}\\&={\left\langle \psi _{W_1},\psi _{W_2}\right\rangle }\cap J_{W_1}[x_{E_2}]\cap J_{W_2}[x_{E_1}], \end{aligned}$$
and hence,
$$\begin{aligned} \Sigma _W=(X_{W_1}\times X_{W_2})\cup (\Sigma _{W_1}\times \mathbb {K}^{E_2})\cup (\mathbb {K}^{E_1}\times \Sigma _{W_2}). \end{aligned}$$
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
(Non-smooth loci and second degeneracy schemes). Let \(W\subseteq \mathbb {K}^E\) be a configuration. Then there is an equality of reduced loci
$$\begin{aligned} \Sigma _W^\text {red}=\Delta _W^\text {red}. \end{aligned}$$
In particular, \(\Sigma _W\) and \(\Delta _W\) have the same generic points, that is,
$$\begin{aligned} {{\,\mathrm{Min}\,}}\Sigma _W={{\,\mathrm{Min}\,}}\Delta _W. \end{aligned}$$
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])
$$\begin{aligned} (Q_W^{i,j})^2=Q_W^{i,j}\cdot Q_W^{j,i}&=Q_W^{i,i}\cdot Q_W^{j,j}-\det Q_W\cdot Q_W^{{\left\{ i,j\right\} },{\left\{ i,j\right\} }}\\&=\partial _i\psi _W\cdot \partial _j\psi _W-\psi _W\cdot Q_W^{{\left\{ i,j\right\} },{\left\{ i,j\right\} }}\in J_W \end{aligned}$$
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
$$\begin{aligned} X_W\cap \mathbb {T}^E&\cong X_{W^\perp }\cap \mathbb {T}^{E^\vee },\\ \Sigma _W\cap \mathbb {T}^E&\cong \Sigma _{W^\perp }\cap \mathbb {T}^{E^\vee },\\ \Delta _W\cap \mathbb {T}^E&\cong \Delta _{W^\perp }\cap \mathbb {T}^{E^\vee }. \end{aligned}$$
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
$$\begin{aligned} {{\,\mathrm{codim}\,}}_{\mathbb {K}^E}\Sigma _W={{\,\mathrm{codim}\,}}_{\mathbb {K}^E}\Delta _W\le 3. \end{aligned}$$
In case of equality, \(\Delta _W\) is Cohen–Macaulay (and hence pure-dimensional) 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 pure-dimensional (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 non-disconnective 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
$$\begin{aligned} \psi _W=\sum _{h\in H}x^{H{\setminus }{\left\{ h\right\} }}\cdot \psi _{W{\setminus } H}+x^H\cdot \psi _{W/H} \end{aligned}$$
has the form (3.14).
-
(a)
Using that \(\psi _W\) is a linear combination of square-free monomials (see Definition 3.2),
$$\begin{aligned} x^{H{\setminus }{\left\{ h\right\} }}\cdot \psi _{W{\setminus } H}=\psi _W\vert _{x_h=0}=\psi _W-x_h\cdot \partial _h\psi _W\in J_W. \end{aligned}$$
-
(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 2-separations). 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) 2-separation of \(\mathsf {M}\). For \({\left\{ i,j\right\} }={\left\{ 1,2\right\} }\), note that
$$\begin{aligned} d_i:=\deg \psi _{W\vert _{E_i}}=\deg \psi _{W/E_j}+1 \end{aligned}$$
and hence by Proposition 3.27
$$\begin{aligned} J_W\ni \psi _W&=\psi _{W/E_i}\cdot \psi _{W\vert _{E_i}}+\psi _{W\vert _{E_j}}\cdot \psi _{W/E_j},\\ J_W\ni \sum _{e\in E_i}x_e\partial _e\psi _W&=d_i\cdot \psi _{W/E_i}\cdot \psi _{W\vert _{E_i}}+(d_i-1)\cdot \psi _{W\vert _{E_j}}\cdot \psi _{W/E_j}. \end{aligned}$$
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 2-separation 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 non-disconnective handle. Suppose that \({{\,\mathrm{codim}\,}}_{\mathbb {K}^{E{\setminus } H}}\Sigma _{W{\setminus } H}=3\). Then \(\Sigma _W\) is equidimensional of codimension
$$\begin{aligned} {{\,\mathrm{codim}\,}}_{\mathbb {K}^E}\Sigma _W=3 \end{aligned}$$
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 non-disconnective, \(\psi _{W{\setminus } H}\in \mathbb {K}[x_{E{\setminus } H}]\) 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 \(d\in H\),
$$\begin{aligned} {{\,\mathrm{height}\,}}({\left\langle J_{W{\setminus } H},x_d\right\rangle })={{\,\mathrm{height}\,}}J_{W{\setminus } H}+1=4. \end{aligned}$$
(4.8)
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),
$$\begin{aligned} x^{H{\setminus }{\left\{ d\right\} }}\cdot \partial _e\psi _{W{\setminus } H}\in \mathfrak {p}+{\left\langle x_d\right\rangle }. \end{aligned}$$
(4.9)
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
$$\begin{aligned} {\left\langle \psi _{W{\setminus } H},\psi _{W/H},x_d,x_h\right\rangle }=\mathfrak {p}_{H,d,h}+{\left\langle \psi _{W/H}\right\rangle },\quad {\left\langle J_{W{\setminus } H},x_d\right\rangle }, \end{aligned}$$
(4.10)
for some \(h\in H{\setminus }{\left\{ d\right\} }\). By Lemma 2.4.(b) and (e) (see Remark 3.5),
$$\begin{aligned} \deg \psi _{W/H}&={{\,\mathrm{rk}\,}}(\mathsf {M}/H)={{\,\mathrm{rk}\,}}\mathsf {M}-{\left| H\right| }\\&={{\,\mathrm{rk}\,}}\mathsf {M}-{{\,\mathrm{rk}\,}}(H)={{\,\mathrm{rk}\,}}(\mathsf {M}{\setminus } H)-\lambda _\mathsf {M}(H)<\deg \psi _{W{\setminus } H} \end{aligned}$$
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 codimension-3 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 pure-dimensional) 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 non-disconnective 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 3-connected matroids). Let \(W\subseteq \mathbb {K}^E\) be a realization of a 3-connected 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,
$$\begin{aligned} {{\,\mathrm{Min}\,}}\Sigma _W={{\,\mathrm{Min}\,}}\Delta _W\subseteq \mathbb {T}^E. \end{aligned}$$
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 1-handle \(H:={\left\{ e\right\} }\in \mathcal {H}_\mathsf {M}\). By Proposition 2.5 and Lemma 2.4.(e), H is non-disconnective 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
$$\begin{aligned} Q_W^{2,3}&=(x_1+x_2)\cdot (-x_3x_5),\\ Q_W^{2,4}&=(x_1+x_2)\cdot (-x_3)\cdot (x_5+x_6),\\ Q_W^{3,4}&=(x_1+x_2)\cdot (x_3+x_4)\cdot x_5,\\ Q_W^{4,4}&=(x_1+x_2)\cdot (x_3+x_4)\cdot (x_5+x_6), \end{aligned}$$
one computes that
$$\begin{aligned} -Q_W^{2,3}+Q_W^{2,4}-Q_W^{3,4}+Q_W^{4,4}=(x_1+x_2)\cdot x_4x_6. \end{aligned}$$
By symmetry, it follows that \(x_2x_4x_6\cdot \mathfrak {p}\subseteq M_W\) and hence
$$\begin{aligned} \Delta _W\cap D(x_2x_4x_6)=V(\mathfrak {p})\cap D(x_2x_4x_6). \end{aligned}$$
Using \(\psi _W\) from Example 3.17, one computes that
$$\begin{aligned}&(x_2\cdot (x_2\partial _2-1)+x_4x_6\cdot (\partial _3+\partial _5)+(x_4+x_6)\cdot (1-x_4\partial _4-x_6\partial _6))\psi _W\\&\quad =2\cdot (x_1+x_2)\cdot x_4^2x_6^2. \end{aligned}$$
By symmetry, it follows that \(2\cdot x_2^2x_4^2x_6^2\cdot \mathfrak {p}\subseteq J_W\) and hence
$$\begin{aligned} \Sigma _W\cap D(x_2x_4x_6)=V(\mathfrak {p})\cap D(x_2x_4x_6). \end{aligned}$$
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
$$\begin{aligned} \bar{I}:=(I+{\left\langle x_e\right\rangle })/{\left\langle x_e\right\rangle }\unlhd \mathbb {K}[x]/{\left\langle x_e\right\rangle }=\mathbb {K}[x_{E{\setminus }{\left\{ e\right\} }}]. \end{aligned}$$
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 non-empty subset \(\gamma (\mathfrak {p})\subseteq {{\,\mathrm{Min}\,}}\Sigma _{W{\setminus } e}\) such that
$$\begin{aligned}&V(\mathfrak {p})\cap V(x_e)=\bigcup _{\mathfrak {q}\in \gamma (\mathfrak {p})}V(\mathfrak {q}), \end{aligned}$$
(4.11)
$$\begin{aligned}&\mathfrak {p}\ne \mathfrak {p}'\implies \gamma (\mathfrak {p})\cap \gamma (\mathfrak {p}')=\emptyset . \end{aligned}$$
(4.12)
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,
$$\begin{aligned} \Sigma _{W{\setminus } e}=\emptyset&\iff J_{W{\setminus } e}=\mathbb {K}[x_{E{\setminus }{\left\{ e\right\} }}] \implies \bar{J}_W=\mathbb {K}[x]/{\left\langle x_e\right\rangle }\\&\iff J_W+{\left\langle x_e\right\rangle }=\mathbb {K}[x] \iff J_W=\mathbb {K}[x] \iff \Sigma _W=\emptyset \end{aligned}$$
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,
$$\begin{aligned} J_{W{\setminus } e}\subseteq \bar{J}_W\subseteq \bar{\mathfrak {q}}. \end{aligned}$$
(4.13)
Since \(x_e\not \in \mathfrak {p}\) by hypothesis, Lemma 4.1 shows that
$$\begin{aligned} {{\,\mathrm{height}\,}}\mathfrak {q}&={{\,\mathrm{height}\,}}\mathfrak {p}+1,\\ {{\,\mathrm{height}\,}}\bar{\mathfrak {q}}&={{\,\mathrm{height}\,}}\mathfrak {q}-{{\,\mathrm{height}\,}}{\left\langle x_e\right\rangle }={{\,\mathrm{height}\,}}\mathfrak {p}. \end{aligned}$$
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 non-empty 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
$$\begin{aligned} R:=\mathbb {K}[\Sigma _W]_\mathfrak {q}. \end{aligned}$$
The inclusion (4.13) gives rise to a surjection of local rings
$$\begin{aligned} \mathbb {K}[\Sigma _{W{\setminus } e}]_{\bar{\mathfrak {q}}}\twoheadrightarrow \mathbb {K}[\Sigma _W\cap V(x_e)]_{\bar{\mathfrak {q}}}=R/tR. \end{aligned}$$
(4.14)
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
$$\begin{aligned} \psi _W&=x^G\cdot \psi _{W/G}+p,\nonumber \\ \partial _f\psi _W&=x^G\cdot \partial _f\psi _{W/G}+\partial _fp,\nonumber \\ Q_W^{i,j}&=x^G\cdot Q_{W/G}^{i,j}+q_{i,j}, \end{aligned}$$
(4.15)
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 square-free 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
$$\begin{aligned} I&:={\left\langle x_F\right\rangle }\unlhd \mathbb {K}[\Sigma _W\cap D(x^G)]=:R\\&=\mathbb {K}[\Sigma _W]_{x^G}=(\mathbb {K}[x]/J_W)_{x^G}=\mathbb {K}[x_F,x_G^{\pm 1}]/(J_W)_{x^G}, \end{aligned}$$
R being equidimensional by hypothesis. With notation from Lemma 4.31
$$\begin{aligned} \bar{R}={{\,\mathrm{gr}\,}}_I R&={{\,\mathrm{gr}\,}}_I((\mathbb {K}[x]/J_W)_{x^G}) \cong ({{\,\mathrm{gr}\,}}_{{\left\langle x_F\right\rangle }}(\mathbb {K}[x]/J_W))_{x^G}\\&\cong (\mathbb {K}[x]/J_W^{\inf })_{x^G} =\mathbb {K}[x_F,x_G^{\pm 1}]/(J_W^{\inf })_{x^G}. \end{aligned}$$
Lemma 4.31 then yields the first claim:
$$\begin{aligned} \Sigma _{W/G}&=\emptyset \iff J_{W/G}=\mathbb {K}[x_F] \iff J_{W/G}[x_G^{\pm 1}]=\mathbb {K}[x_F,x_G^{\pm 1}]\\&\implies (J_W^{\inf })_{x^G}=\mathbb {K}[x_F,x_G^{\pm 1}] \iff \bar{R}=0\iff I=R\\&\iff \Sigma _W\cap D(x^G)\cap V(x_F)=\emptyset . \end{aligned}$$
The latter equality makes the second claim vacuous.
We may thus assume that \(I\ne R\). Lemma 4.31 yields a surjection
$$\begin{aligned} \pi :\mathbb {K}[\Sigma _{W/G}\times \mathbb {T}^G]&=(\mathbb {K}[x_F]/J_{W/G})[x_G^{\pm 1}]\\&=\mathbb {K}[x_F,x_G^{\pm 1}]/(J_{W/G}[x_G^{\pm 1}])\twoheadrightarrow \bar{R}. \end{aligned}$$
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
$$\begin{aligned} {{\,\mathrm{Min}\,}}{{\,\mathrm{Spec}\,}}\bar{R}\subseteq {{\,\mathrm{Min}\,}}(\Sigma _{W/G}\times \mathbb {T}^G). \end{aligned}$$
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
$$\begin{aligned} \pi _\mathfrak {p}:\mathbb {K}[\Sigma _{W/G}\times \mathbb {T}^G]_\mathfrak {p}\twoheadrightarrow \bar{R}_\mathfrak {p}\end{aligned}$$
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 non-maximal 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
$$\begin{aligned} \hbar :={\left| E\right| }-{\left| {{\,\mathrm{Max}\,}}\mathcal {H}_\mathsf {M}\right| }\ge 0. \end{aligned}$$
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)
$$\begin{aligned} {{\,\mathrm{height}\,}}(\mathfrak {p}+{\left\langle x_F\right\rangle }) \le 3+{\left| F\right| } =3+{\left| {{\,\mathrm{Max}\,}}\mathcal {H}_\mathsf {M}\right| } <{\left| E\right| }={{\,\mathrm{height}\,}}{\left\langle x\right\rangle }. \end{aligned}$$
(4.16)
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
$$\begin{aligned} \mathfrak {p}+{\left\langle x_F\right\rangle }\subseteq \mathfrak {q}\subsetneq {\left\langle x\right\rangle }. \end{aligned}$$
(4.17)
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,
$$\begin{aligned} \mathfrak {p}\in D(x^G),\quad \mathfrak {q}\in V(\mathfrak {p})\cap D(x^G)\cap V(x_F)\ne \emptyset . \end{aligned}$$
(4.18)
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 pure-dimensional. 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) non-disconnective handles \(H=H_1,\dots ,H_\ell \in \mathcal {H}_\mathsf {M}\) with
$$\begin{aligned} \ell \ge k+1\ge 3. \end{aligned}$$
(4.19)
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)
$$\begin{aligned} m:={\left| {{\,\mathrm{Max}\,}}\mathcal {H}_\mathsf {M}\right| },\quad \hbar :={\left| E\right| }-m. \end{aligned}$$
If \(\hbar >3\), then \(\Sigma _W\) is generically reduced by Lemma 4.34.(a) and the induction hypothesis. Otherwise,
$$\begin{aligned} 2\ell +(m-\ell )\le \sum _{i=1}^\ell {\left| H_i\right| }+(m-\ell )\le {\left| E\right| }=\hbar +m\le 3+m \end{aligned}$$
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, \(\hbar =6-3=3>2\). 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
$$\begin{aligned} \Delta _W=\Sigma _W^\text {red}\end{aligned}$$
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 3-connected matroids). Let \(W\subseteq \mathbb {K}^E\) be a realization of a 3-connected 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 3-connectedness 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
$$\begin{aligned} \mathbb {T}^E\supseteq {{\,\mathrm{Min}\,}}\Delta _W={{\,\mathrm{Min}\,}}\Delta _{W^\perp }\subseteq \mathbb {T}^{E^\vee }. \end{aligned}$$
(4.20)
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 wheels-and-whirls theorem (see [26, Thm. 8.8.4]) yields an \(e\in E\) such that \(\mathsf {M}{\setminus } e\) or \(\mathsf {M}/e\) is again 3-connected. In the latter case, we replace W by \(W^\perp \) and use (2.11). We may thus assume that \(\mathsf {M}{\setminus } e\) is 3-connected. 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
$$\begin{aligned} s_i\mapsto s_{i+1},\quad r_i\mapsto r_{i+1},\quad w^i\mapsto w^{i+1}. \end{aligned}$$
(4.21)
Proof
By Lemma 2.27, W has a basis \(w=(w_1,\dots ,w_n)\) where \(w^i=s_i+r_i-r_{i-1}\) 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 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.
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
$$\begin{aligned} \psi _W=\det Q_n,\quad M_W=I_{n-1}(Q_n), \end{aligned}$$
where
$$\begin{aligned} Q_n:= \begin{pmatrix} z'_1 &{} \quad y_1 &{} \quad 0 &{} \quad \cdots &{} \quad \cdots &{} \quad 0 &{} \quad y_n\\ y_1 &{} \quad z'_2 &{} \quad y_2 &{} \quad 0 &{} \quad \cdots &{} \quad \cdots &{} \quad 0 \\ 0 &{} \quad y_2 &{} \quad z'_3 &{} \quad y_3 &{} \quad 0 &{} \quad \cdots &{} \quad 0\\ \vdots &{} \quad \ddots &{} \quad \ddots &{} \quad \ddots &{} \quad \ddots &{} \quad \ddots &{} \quad \vdots \\ 0 &{} \quad \cdots &{} \quad 0 &{} \quad y_{n-3} &{} \quad z'_{n-2} &{} \quad y_{n-2} &{} \quad 0\\ 0 &{} \quad \cdots &{} \quad \cdots &{} \quad 0 &{} \quad y_{n-2} &{} \quad z'_{n-1} &{} \quad y_{n-1} \\ y_n &{} \quad 0 &{} \quad \cdots &{} \quad \cdots &{} \quad 0 &{} \quad y_{n-1} &{} \quad z'_n \end{pmatrix}. \end{aligned}$$
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
$$\begin{aligned} z_1,\dots ,z_n,y_1,\dots ,y_n:=s_1^\vee ,\dots ,s_n^\vee ,r_1^\vee ,\dots ,r_n^\vee , \end{aligned}$$
(4.22)
and consider the \(\mathbb {K}\)-linear automorphism defined by
$$\begin{aligned} z'_1:=z_1+y_1+t^2\cdot y_n,\quad z'_i:=z_i+y_i+y_{i-1},\quad i=2,\dots ,n. \end{aligned}$$
Then \(Q_W\) is represented by the matrix
$$\begin{aligned} \begin{pmatrix} z'_1 &{} \quad -y_1 &{} \quad 0 &{} \quad \cdots &{} \quad \cdots &{} \quad 0 &{} \quad -t\cdot y_n\\ -y_1 &{} \quad z'_2 &{} \quad -y_2 &{} \quad 0 &{} \quad \cdots &{} \quad \cdots &{} \quad 0 \\ 0 &{} \quad -y_2 &{} \quad z'_3 &{} \quad -y_3 &{} \quad 0 &{} \quad \cdots &{} \quad 0\\ \vdots &{} \quad \ddots &{} \quad \ddots &{} \quad \ddots &{} \quad \ddots &{} \quad \ddots &{} \quad \vdots \\ 0 &{} \quad \cdots &{} \quad 0 &{} \quad -y_{n-3} &{} \quad z'_{n-2} &{} \quad -y_{n-2} &{} \quad 0\\ 0 &{} \quad \cdots &{} \quad \cdots &{} \quad 0 &{} \quad -y_{n-2} &{} \quad z'_{n-1} &{} \quad -y_{n-1} \\ -t\cdot y_n &{} \quad 0 &{} \quad \cdots &{} \quad \cdots &{} \quad 0 &{} \quad -y_{n-1} &{} \quad z'_n \end{pmatrix}. \end{aligned}$$
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 \(k=n-2\).
Consider the morphism of algebraic varieties of matrices
$$\begin{aligned} Y:=\mathbb {K}^{n\times k}\rightarrow {\left\{ A\in \mathbb {K}^{n\times n}\mid A=A^t,\ {{\,\mathrm{rk}\,}}A\le k\right\} }=:Z,\quad B\mapsto BB^t. \end{aligned}$$
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
$$\begin{aligned} V(y_{1,1}y_{1,3}+y_{1,2}y_{2,3},y_{2,1}y_{1,4}+y_{2,2}y_{2,4}) \end{aligned}$$
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 non-disconnective handles, the 2-handle \({\left\{ r_{n-1},r_n\right\} }\) and 1-handles.
-
(c)
The matroid \(\mathsf {M}/r_n\) is connected of rank \({{\,\mathrm{rk}\,}}(\mathsf {M}/r_n)\ge 2\). Its handle partition consists of non-disconnective 1-handles.
-
(d)
We can identify \(\mathsf {W}_n{\setminus } s_n/r_n=\mathsf {W}_{n-1}\) and \(\mathsf {W}^n{\setminus } s_n/r_n=\mathsf {W}^{n-1}\).
Proof
-
(a)
The self-duality 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 \({\left\{ s_{n-1},r_{n-1},s_n\right\} }\) and maps the triangle \({\left\{ s_{n},r_{n},s_1\right\} }\) to \({\left\{ s_{n-1},r_{n-1},s_1\right\} }\) (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
$$\begin{aligned} \mathsf {M}/r_n{\setminus } s_n=\mathsf {M}\setminus s_n/r_n=\mathsf {M}_{n-1} \end{aligned}$$
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 \(V(y_{n-1},y_n)\) 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
$$\begin{aligned} (\mathsf {M}{\setminus } s_n)^\perp =\mathsf {M}^\perp /s_n^\vee =\mathsf {M}/r_n,\quad E^\vee {\setminus }{\left\{ s_n^\vee \right\} }=E{\setminus }{\left\{ r_n\right\} }, \end{aligned}$$
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,
$$\begin{aligned} {{\,\mathrm{Min}\,}}\Delta _{W{\setminus } s_n}={\left\{ \mathfrak {q},\mathfrak {q}'\right\} },\quad \mathfrak {q}\in \mathbb {T}^{E{\setminus }{\left\{ s_n\right\} }},\quad \mathfrak {q}'\in V(y_{n-1},y_n). \end{aligned}$$
(4.23)
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
$$\begin{aligned} \Delta _W={\left\{ \mathfrak {p},\mathfrak {p}'\right\} }. \end{aligned}$$
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 \(\sqrt{\mathfrak {p}'+{\left\langle z_i\right\rangle }}\supseteq {\left\langle y_{i-1},y_i\right\rangle }\) for all \(i=1,\dots ,n\) and hence
$$\begin{aligned} \sqrt{\mathfrak {p}'+{\left\langle z_1,\dots ,z_n\right\rangle }}={\left\langle z_1,\dots ,z_n,y_1,\dots ,y_n\right\rangle }. \end{aligned}$$
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 \(\sqrt{\mathfrak {p}+{\left\langle z_{2i}\right\rangle }}\supseteq {\left\langle y_{2i-1},y_{2i}\right\rangle }\) for all \(i=1,\dots ,m\) and hence
$$\begin{aligned} \sqrt{\mathfrak {p}+{\left\langle z_2,z_4,z_6,\dots ,z_n\right\rangle }}\supseteq {\left\langle z_2,z_4,z_6,\dots ,z_n,y_1,\dots ,y_n\right\rangle }. \end{aligned}$$
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 non-adjacent spokes \(s_i\) and \(s_j\). Then
$$\begin{aligned} \sqrt{\mathfrak {p}+{\left\langle z_i,z_j\right\rangle }}\supseteq {\left\langle z_i,z_j,y_{i-1},y_i,y_{j-1},y_j\right\rangle }. \end{aligned}$$
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 3-connectedness). Let \(\mathsf {M}\) be a matroid of rank \({{\,\mathrm{rk}\,}}\mathsf {M}\ge 2\) on E. Then \(\mathsf {M}\) is 3-connected 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.