1 Introduction

Let \(\mathbb{S}^n\) be the space of symmetric \(n\times n\) matrices over the complex numbers \({{{\mathbb {C}}\,}}\). We say that a subspace \({{{\mathcal{L}}\,}}\subseteq \mathbb{S}^n\) is regular when it contains an invertible matrix. For a regular subspace \({{{\mathcal{L}}\,}}\subseteq \mathbb{S}^n\), we define its reciprocal variety \({{{\mathcal{L}}\,}}^{-1}\) to be the Zariski closure of the set \(\{X^{-1}\mid X\in {{{\mathcal{L}}\,}},\det (X)\ne 0\}\). The goal of this paper is to follow up on [1] and study the geometry of the regular subspaces \({{{\mathcal{L}}\,}}\subseteq \mathbb{S}^n\) such that \({{{\mathcal{L}}\,}}^{-1}\) is again a linear subspace of \(\mathbb{S}^n\). Denote the Grassmannian of m-dimensional subspaces of \(\mathbb{S}^n\) by \({{\mathrm{Gr}\,}}(m,\mathbb{S}^n)\). The following theorem, which is a more general formulation of a result of Jensen [7, Lemma 1], was the starting point of our investigation.

Theorem 1.1

[1, Theorem 1.1] Let \({{{\mathcal{L}}\,}}\in {{\mathrm{Gr}\,}}(m,\mathbb{S}^n)\) be a regular subspace and let \(U\in {{{\mathcal{L}}\,}}\) be an invertible matrix. Then the following are equivalent:

\(\mathrm {(a)}\):

The reciprocal variety \({{{\mathcal{L}}\,}}^{-1}\) is also a linear space in \(\mathbb{S}^n\).

\(\mathrm {(b)}\):

The subspace \({{{\mathcal{L}}\,}}\) is a subalgebra of the Jordan algebra \((\mathbb{S}^n,\bullet _U)\).

\(\mathrm {(c)}\):

We have \({{{\mathcal{L}}\,}}^{-1}=U^{-1}{{{\mathcal{L}}\,}}U^{-1}\).

Here the operation \(-\bullet _U-:\mathbb{S}^n\times \mathbb{S}^n\rightarrow \mathbb{S}^n\) is defined as

$$\begin{aligned} X\bullet _U Y:= \frac{XU^{-1}Y+YU^{-1}X}{2}\in \mathbb{S}^n \end{aligned}$$

for all \(X,Y\in \mathbb{S}^n\) and satisfies the Jordan axiom:

$$\begin{aligned} (X\bullet _U X)\bullet _U(X\bullet _U Y)=X\bullet _U((X\bullet _UX)\bullet _U Y)\hbox { for all}\; X,Y\in \mathbb{S}^n. \end{aligned}$$

We call a subspace \({{{\mathcal{L}}\,}}\in {{\mathrm{Gr}\,}}(m,\mathbb{S}^n)\) a Jordan space when these equivalent conditions are satisfied. Jordan spaces of dimensions 2, 3 and 4 are also called Jordan pencils, nets and webs, respectively.

Denote the adjoint of a matrix \(U\in \mathbb{S}^n\) by \({{\mathrm{adj}\,}}(U)\). Define the Jordan locus \({{\mathrm{Jo}\,}}(m,\mathbb{S}^n)\) to be the subvariety of \({{\mathrm{Gr}\,}}(m,\mathbb{S}^n)\) consisting of all subspaces \({{{\mathcal{L}}\,}}={{{\mathbb {C}}\,}}\{X_1,\ldots ,X_m\}\) such that

$$\begin{aligned} X_1,X_2,\ldots ,X_m, \,\,X_i{{\mathrm{adj}\,}}(U)X_j+X_j{{\mathrm{adj}\,}}(U)X_i \end{aligned}$$

are linearly dependent for all matrices \(U\in {{{\mathcal{L}}\,}}\) and all indices \(1\le i\le j\le m\). Theorem 1.1 shows that a regular subspace \({{{\mathcal{L}}\,}}\in {{\mathrm{Gr}\,}}(m,\mathbb{S}^n)\) is a Jordan space if and only if \({{{\mathcal{L}}\,}}\in {{\mathrm{Jo}\,}}(m,\mathbb{S}^n)\). The group \({{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}})\) acts on \({{\mathrm{Gr}\,}}(m,\mathbb{S}^n)\) by congruence:

$$\begin{aligned} g\cdot {{{\mathcal{L}}\,}}:=g{{{\mathcal{L}}\,}}g^\top \in {{\mathrm{Gr}\,}}(m,\mathbb{S}^n)\hbox { for all}\; {{{\mathcal{L}}\,}}\in {{\mathrm{Gr}\,}}(m,\mathbb{S}^n) \hbox {and}\; g\; \in {{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}}). \end{aligned}$$

The subvariety \({{\mathrm{Jo}\,}}(m,\mathbb{S}^n)\) is closed under the congruence action. It is the goal of this paper to classify the orbits \({{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}})\cdot {{{\mathcal{L}}\,}}\) of regular subspaces \({{{\mathcal{L}}\,}}\in {{\mathrm{Jo}\,}}(m,S^n)\) and to understand the degenerations between them.

1.1 Results

1.1.1 Classification of orbits

For 5 out of 6 isomorphism types of unital Jordan algebras of dimension 3, we determine the congruence orbits of Jordan nets in \(\mathbb{S}^n\) for all n. For the last isomorphism type, we do this for \(n\le 7\). We also determine the congruence orbits of embeddings of Jordan webs in \(\mathbb{S}^n\) for \(n\le 5\). These results are summarized in Sect. 3.

1.1.2 Degenerations and obstructions

In Appendix, we give lists of degenerations between orbits of Jordan spaces and in Sect. 6 we list a series of obstructions to the existence of such degenerations. These obstructions suffice to show that our lists of degenerations Jordan nets of the first 5 isomorphism types are complete. We also determine the degeneration diagrams of Jordan nets in \(\mathbb{S}^5\) and \(\mathbb{S}^6\) (up to one degeneration) in Sect. 4.

1.1.3 Numerical results

We give an algorithm that indicates numerically whether a degeneration between two orbits of Jordan spaces exists. We verified this algorithm using all known degenerations and obstructions between Jordan nets in \(\mathbb{S}^n\) for \(n\le 6\) and use the algorithm to compute the degenerations between Jordan nets in \(\mathbb{S}^7\) and Jordan webs in \(\mathbb{S}^n\) for \(n=4,5\).

1.2 Applications

Subspaces \({{{\mathcal{L}}\,}}\subseteq \mathbb{S}^n\) such that \({{{\mathcal{L}}\,}}^{-1}\) is also a linear spaces arise naturally in statistics [7, 15]: In many statistical applications, one studies normally distributed random variables \(X_1,\ldots ,X_n\) with linear conditions on their covariance matrix \(\Sigma\) or their concentration matrix \(\Sigma ^{-1}\). The condition that the matrix \(\Sigma\) lies in a subspace \({{{\mathcal{L}}\,}}\subseteq \mathbb{S}^n\) whose reciprocal is also a linear space is a mixture of these two conditions. Seely [13, 14] proved that such models are the only models of multivariate normal distributions with zero mean that have a complete sufficient statistic.

In [12], Parrilo and Permenter showed that minimal subspaces which contain primal and dual solutions of a semidefinite optimization problem are Jordan algebras. Indeed, Jordan algebras are in some sense the more general space for optimization problems. It is well known that linear optimization problems are equivalent to semidefinite optimization problems over diagonal matrices; however, both are instances of symmetric cones [5]. Symmetric cones are given by the squares of an Euclidean Jordan algebra, i.e., a Jordan algebra where sum of squares are always nonzero.

1.3 Structure of the paper

In Sect. 2, we recall the basic properties of abstract unital Jordan algebras and list the degenerations between them in dimensions 2, 3 and 4. In Sect. 3, we list the orbits of embeddings of Jordan nets into \(\mathbb{S}^n\) for \(n\le 7\) and Jordan webs into \(\mathbb{S}^n\) for \(n\le 5\). In Sect. 4, we list all degenerations of Jordan nets in \(\mathbb{S}^n\) for \(n=5,6\) assuming one degeneration does not exist. In Sect. 5, we find all degenerations between Jordan nets in \(\mathbb{S}^7\) and Jordan webs in \(\mathbb{S}^n\) for \(n=4,5\) numerically. In Sect. 6, we give a list of obstructions to the existence of a degeneration between two orbits. In Sect. 7, we find all degenerations between some families of Jordan nets. In Sect. 8, we prove the results from Sect. 4. In Appendix A, we classify embeddings of indecomposable Jordan algebras into \(\mathbb{S}^n\) and prove the results from Sect. 3. In Appendix B, we list some families of degenerations between Jordan algebras.

2 Abstract Jordan algebras

The goal of this section is to review the basic notions concerning (abstract unital) Jordan algebras. For Jordan algebras of dimension \(\le 4\), we give their classification up to isomorphism and the degenerations between the different isomorphism classes.

Definition 2.1

An (abstract) Jordan algebra \({{\mathrm{\mathcal{A}}\,}}\) is a complex vector space equipped with a symmetric bilinear operation \(-\cdot -:{{\mathrm{\mathcal{A}}\,}}\times {{\mathrm{\mathcal{A}}\,}}\rightarrow {{\mathrm{\mathcal{A}}\,}}\) such that

$$\begin{aligned} (x\cdot x)\cdot (x\cdot y)=x\cdot ((x\cdot x)\cdot y) \end{aligned}$$

holds for all \(x,y\in {{\mathrm{\mathcal{A}}\,}}\). This condition is called the Jordan axiom. A Jordan algebra \({{\mathrm{\mathcal{A}}\,}}\) is called unital if there exists an \(u\in {{\mathrm{\mathcal{A}}\,}}\) such that \(u\cdot x=x\) for all \(x\in {{\mathrm{\mathcal{A}}\,}}\).‚ô¶

All Jordan algebras in this paper are assumed to be unital. We often write xy instead of \(x\cdot y\) and \(x^d\) instead of \(x\cdot x^{d-1}\) for \(d\ge 2\).

Example 2.2

For any invertible matrix \(U\in \mathbb{S}^n\), the operation

$$\begin{aligned} -\bullet _U-:{{\mathrm{\mathcal{A}}\,}}\times {{\mathrm{\mathcal{A}}\,}}\rightarrow & {} {{\mathrm{\mathcal{A}}\,}}\\ (X,Y)\mapsto & {} \frac{1}{2}\left( XU^{-1}Y+YU^{-1}X\right) \end{aligned}$$

defines a Jordan algebra structure on \(\mathbb{S}^n\) where the matrix U is the unit.‚ô†

A morphism of Jordan algebras \(\varphi :{{\mathrm{\mathcal{A}}\,}}\rightarrow {{\mathrm{\mathcal{B}}\,}}\) is a linear map that sends the unit of \({{\mathrm{\mathcal{A}}\,}}\) to the unit of \({{\mathrm{\mathcal{B}}\,}}\) such that \(\varphi (x\cdot y)=\varphi (x)\cdot \varphi (y)\) for all \(x,y\in {{\mathrm{\mathcal{A}}\,}}\). An isomorphism is a morphism which is invertible as a linear map.

Example 2.3

Let \(d\ge 1\) and \(0\le r\le d\) be integers and let \(\beta :{{{\mathbb {C}}\,}}^d\times {{{\mathbb {C}}\,}}^d\rightarrow {{{\mathbb {C}}\,}}\) be a symmetric bilinear form of rank r. Define \({{\mathrm{\mathcal{J}}\,}}^d_{r,\beta }\) to be the vector space \({{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}^d\) equipped with the operation

$$\begin{aligned} (\lambda ,v)\cdot (\mu ,w):=(\lambda \mu +\beta (v,w),\lambda w+\mu v). \end{aligned}$$

Then \({{\mathrm{\mathcal{J}}\,}}^d_{r,\beta }\) is a Jordan algebra. One can check that the isomorphism type of \({{\mathrm{\mathcal{J}}\,}}^d_{r,\beta }\) only depends on (d, r). When \(\beta (v,w)=v_1w_1+\cdots +v_rw_r\), we also denote \({{\mathrm{\mathcal{J}}\,}}^d_{r,\beta }\) by \({{\mathrm{\mathcal{J}}\,}}^d_r\).♠

Let \({{\mathrm{\mathcal{A}}\,}}\) be a Jordan algebra with unit u.

Definition 2.4

The rank of \({{\mathrm{\mathcal{A}}\,}}\) is the minimal number \({{\mathrm{rk}\,}}(A):=k\ge 1\) such that \(u,x,x^2,\ldots ,x^k\) are linearly dependent for all \(x\in {{\mathrm{\mathcal{A}}\,}}\).‚ô¶

Example 2.5

Let \(d\ge 1\) and \(0\le r\le d\) again be integers and consider the Jordan algebra \({{\mathrm{\mathcal{J}}\,}}^d_r\). It has unit \(u=(1,0)\). Let \(x=(\lambda ,v)\in {{\mathrm{\mathcal{A}}\,}}\) be any element. Then

$$\begin{aligned} {{\mathrm{span}\,}}(u,x,x^2,\ldots ,x^k)={{\mathrm{span}\,}}(u,x-\lambda u,(x-\lambda u)^2,\ldots ,(x-\lambda u)^k) \end{aligned}$$

for each \(k\ge 1\). So for the purposes of determining the rank of \({{\mathrm{\mathcal{J}}\,}}^d_r\), it suffices to consider the case where \(\lambda =0\). We see that \(u=(1,0)\) and \(x=(0,v)\) are linearly independent in general and that \(u=(1,0)\) and \(x^2=(\beta (v,v),0)\) are linearly dependent for all \(v\in {{{\mathbb {C}}\,}}^d\). Hence \({{\mathrm{rk}\,}}({{\mathrm{\mathcal{J}}\,}}^d_r)=2\).‚ô†

Definition 2.6

We say that \({{\mathrm{\mathcal{A}}\,}}\) is decomposable when \({{\mathrm{\mathcal{A}}\,}}\cong {{\mathrm{\mathcal{B}}\,}}_1\times {{\mathrm{\mathcal{B}}\,}}_2\) where \({{\mathrm{\mathcal{B}}\,}}_1,{{\mathrm{\mathcal{B}}\,}}_2\) are Jordan algebras and \((a_1,a_2)\cdot (b_1,b_2)=(a_1b_1,a_2b_2)\) for all \(a_1,b_1\in {{\mathrm{\mathcal{B}}\,}}_1\) and \(a_2,b_2\in {{\mathrm{\mathcal{B}}\,}}_2\).‚ô¶

Proposition 2.7

Let \({{\mathrm{\mathcal{B}}\,}}_1,{{\mathrm{\mathcal{B}}\,}}_2\) be Jordan algebras with units \(u_1,u_2\) and suppose that \({{\mathrm{\mathcal{A}}\,}}\cong {{\mathrm{\mathcal{B}}\,}}_1\times {{\mathrm{\mathcal{B}}\,}}_2\). Then

$$\begin{aligned} {{\mathrm{rk}\,}}(A)={{\mathrm{rk}\,}}({{\mathrm{\mathcal{B}}\,}}_1)+{{\mathrm{rk}\,}}({{\mathrm{\mathcal{B}}\,}}_2). \end{aligned}$$

Proof

Let \(x\in {{\mathrm{\mathcal{A}}\,}}\) be an element and \(P(X)=a_0+a_1X+\cdots +a_dX^d\in {{{\mathbb {C}}\,}}[X]\) a polynomial. Then we write \(P(x):= a_0u+a_1x+\cdots +a_dx^d\). The minimal polynomial \(P_x\) of x is the monic generator of the ideal \(\{P\in {{{\mathbb {C}}\,}}[X]\mid P(x)=0\}\) of \({{{\mathbb {C}}\,}}[X]\). Note that

$$\begin{aligned} {{\mathrm{rk}\,}}({{\mathrm{\mathcal{A}}\,}})=\max _{x\in {{\mathrm{\mathcal{A}}\,}}}(\deg P_x). \end{aligned}$$

Now let \(b_1,b_2\) be elements of \({{\mathrm{\mathcal{B}}\,}}_1,{{\mathrm{\mathcal{B}}\,}}_2\). Then \(P_{(b_1,b_2)}={{\mathrm{lcm}\,}}(P_{b_1},P_{b_2})\) and so \({{\mathrm{rk}\,}}({{\mathrm{\mathcal{A}}\,}})\le {{\mathrm{rk}\,}}({{\mathrm{\mathcal{B}}\,}}_1)+{{\mathrm{rk}\,}}({{\mathrm{\mathcal{B}}\,}}_2)\). Assume that \({{\mathrm{rk}\,}}({{\mathrm{\mathcal{B}}\,}}_i)=\deg P_{b_i}\). Then there exists a \(\lambda \in {{{\mathbb {C}}\,}}\) such that \(P_{b_1}, P_{b_2}(X-\lambda )=P_{b_2+\lambda u_2}\) have distinct roots. It follows that

$$\begin{aligned} {{\mathrm{rk}\,}}({{\mathrm{\mathcal{A}}\,}})\ge \deg P_{(b_1,b_2+\lambda u_2)}=\deg {{\mathrm{lcm}\,}}(P_{b_1},P_{b_2}(X-\lambda ))=\deg P_{b_1}+\deg P_{b_2}={{\mathrm{rk}\,}}({{\mathrm{\mathcal{B}}\,}}_1)+{{\mathrm{rk}\,}}({{\mathrm{\mathcal{B}}\,}}_2) \end{aligned}$$

and hence \({{\mathrm{rk}\,}}({{\mathrm{\mathcal{A}}\,}})={{\mathrm{rk}\,}}({{\mathrm{\mathcal{B}}\,}}_1)+{{\mathrm{rk}\,}}({{\mathrm{\mathcal{B}}\,}}_2)\). \(\square\)

Let \(n\ge 1\) be an fixed integer and let \(x_1,\ldots ,x_n\) be a basis of a Jordan algebra \({{\mathrm{\mathcal{A}}\,}}\). Then the Jordan algebra structure is determined by the constants \(c_i^{j_1,j_2}\in {{{\mathbb {C}}\,}}\) such that

$$\begin{aligned} x_{j_1}\cdot x_{j_2}= \sum _{i=1}^k c_i^{j_1,j_2}\cdot x_i. \end{aligned}$$

We define the space \({{\mathrm{Jord}\,}}_n\) of abstract Jordan algebras of dimension n to be the subvariety of \({{{\mathbb {C}}\,}}^{n\times n\times n}\) consisting of all elements \((c_i^{j_1,j_2})_{i,j_1,j_2}\) defining a Jordan algebra structure. This means that \(c_i^{j_1,j_2}=c_i^{j_2,j_1}\) and that the Jordan axiom is satisfied. Note that \({{\mathrm{GL}\,}}_n\) acts on \({{\mathrm{Jord}\,}}_n\) via base change. The orbit of a Jordan algebra \({{\mathrm{\mathcal{A}}\,}}\) consists of all Jordan algebras isomorphic to \({{\mathrm{\mathcal{A}}\,}}\).

Definition 2.8

We say that a Jordan algebra \({{\mathrm{\mathcal{A}}\,}}\) degenerates to a Jordan algebra \({{\mathrm{\mathcal{B}}\,}}\) when there exists a matrix \(g(t)\in {{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}}[t^{\pm 1}])\) such that \({{\mathrm{\mathcal{B}}\,}}=\lim _{t\rightarrow 0}g(t)\cdot {{\mathrm{\mathcal{A}}\,}}\). We denote this by \({{\mathrm{\mathcal{A}}\,}}\rightarrow {{\mathrm{\mathcal{B}}\,}}\). Equivalently, we say that \({{\mathrm{\mathcal{A}}\,}}\) degenerates to \({{\mathrm{\mathcal{B}}\,}}\) (topologically) when \({{\mathrm{\mathcal{B}}\,}}\) is in the orbit-closure of \({{\mathrm{\mathcal{A}}\,}}\).‚ô¶

For a proof that these definitions are equivalent, see the proof of [3, Theorem 20.24].

Example 2.9

There are two isomorphism classes of 2-dimensional Jordan algebras, namely \({{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\) and \({{\mathrm{\mathcal{J}}\,}}^1_0\). The former has a basis x, y with \(x^2=x\), \(xy=0\) and \(y^2=y\). The latter has a basis u, z where u is the unit and \(z^2=0\). For \(t\ne 0\), the basis \((u,z_t)=(x+y,ty)\) satisfies

$$\begin{aligned} u^2&= 1\cdot u+0\cdot z_t,\\ uz_t&= 0\cdot u+1\cdot z_t, \\ z_t^2&= 0\cdot u+t\cdot z_t \end{aligned}$$

Taking the limit of the structure constants (1, 0, 0, 1, 0, t) for \(t\rightarrow 0\), we find the structure constants (1, 0, 0, 1, 0, 0) of \({{\mathrm{\mathcal{J}}\,}}^1_0\). Hence \({{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\rightarrow {{\mathrm{\mathcal{J}}\,}}^1_0\).♠

The Jordan algebras of dimension 4 were classified by Martin.

Theorem 2.10

(Martin [10]) Let \({{\mathrm{\mathcal{A}}\,}}\) be an indecomposable unital Jordan algebra of dimension \(\le 4\). Then \({{\mathrm{\mathcal{A}}\,}}\) is isomorphic to one of the following Jordan algebras:

  1. (1)

    \({{{\mathbb {C}}\,}}\)

  2. (2)

    \({{\mathrm{\mathcal{J}}\,}}_0^1\)

  3. (3)

    \({{\mathrm{\mathcal{J}}\,}}_0^2,{{\mathrm{\mathcal{J}}\,}}_1^2,{{\mathrm{\mathcal{J}}\,}}_2^2,{{{\mathbb {C}}\,}}[x]/(x^3)\)

  4. (4)

    \({{\mathrm{\mathcal{J}}\,}}_0^3,{{\mathrm{\mathcal{J}}\,}}_1^3,{{\mathrm{\mathcal{J}}\,}}_2^3,{{\mathrm{\mathcal{J}}\,}}_3^3,{{{\mathbb {C}}\,}}[x]/(x^4)\), the subalgebras

    $$\begin{aligned} {{\mathrm{\mathcal{E}}\,}}_1:=\begin{pmatrix}v&{}w&{}x\\ w&{}y\\ x\end{pmatrix},{{\mathrm{\mathcal{E}}\,}}_2:=\begin{pmatrix}v&{}x&{}w\\ x\\ w&{}&{}&{}y\\ &{}&{}y\end{pmatrix},{{\mathrm{\mathcal{E}}\,}}_3:=\begin{pmatrix}y&{}x&{}u&{}z\\ x&{}u\\ u\\ z&{}&{}&{}u\end{pmatrix} \end{aligned}$$

    of \(\mathbb{S}^3,\mathbb{S}^4\) and \({{\mathrm{\mathcal{E}}\,}}_4:={{{\mathbb {C}}\,}}[x,y]/(x^2,xy,y^2)\).

We know the degeneration diagrams for Jordan algebras of dimension 3 and 4.

Theorem 2.11

The following diagram gives all degenerations between 3-dimensional Jordan algebras.

figure a

To prove the theorem, we need the following proposition.

Proposition 2.12

If \({{\mathrm{\mathcal{A}}\,}}\rightarrow {{\mathrm{\mathcal{B}}\,}}\), then \({{\mathrm{rk}\,}}(B)\le {{\mathrm{rk}\,}}({{\mathrm{\mathcal{A}}\,}})\).

Proof

This follows from the fact that bounded rank is a closed condition. \(\square\)

Proof of Theorem 2.11

The degenerations and the fact that \({{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\not \rightarrow {{\mathrm{\mathcal{J}}\,}}^2_1\) are obtained by [11] and based on partial results from [9]. We have \({{\mathrm{rk}\,}}({{\mathrm{\mathcal{J}}\,}}^2_2)=2<3={{\mathrm{rk}\,}}({{{\mathbb {C}}\,}}[x]/(x^3))\) and hence \({{\mathrm{\mathcal{J}}\,}}_2^2\not \rightarrow {{{\mathbb {C}}\,}}[x]/(x^3)\) by the previous proposition. \(\square\)

Theorem 2.13

(Martin [11], Kashuba-Martin [8]) The following diagram gives all degenerations between 4-dimensional Jordan algebras.

figure b

3 Jordan nets in \(\mathbb{S}^n\) for \(n\le 7\) and Jordan webs in \(\mathbb{S}^n\) for \(n\le 5\)

Definition 3.1

Let \(n\ge 1\) be an integer. An embedding of a Jordan algebra \({{\mathrm{\mathcal{A}}\,}}\) into \(\mathbb{S}^n\) is the image of an injective morphism of Jordan algebras \({{\mathrm{\mathcal{A}}\,}}\rightarrow \mathbb{S}^n\), where \(\mathbb{S}^n\) is equipped with the product \(\bullet _U\) for any invertible \(U\in \mathbb{S}^n\). A subspace \({{{\mathcal{L}}\,}}\subseteq \mathbb{S}^n\) is called a Jordan space if it is the embedding of some Jordan algebra into \(\mathbb{S}^n\). We denote the set of m-dimensional Jordan spaces in \(\mathbb{S}^n\) by \({{\mathrm{Jo}\,}}(m,\mathbb{S}^n)\). We denote the subset of \({{\mathrm{Jo}\,}}(m,\mathbb{S}^n)\) of subspaces containing \(\mathbf{1}_n\) by \({{\mathrm{Jo}\,}}_\mathbf{1}(m,\mathbb{S}^n)\).‚ô¶

Jordan spaces of dimensions 2, 3 and 4 are also called Jordan pencils, nets and webs, respectively.

Definition 3.2

Two subspaces \({{{\mathcal{L}}\,}},{{{\mathcal{L}}\,}}'\subseteq \mathbb{S}^n\) are congruent if \({{{\mathcal{L}}\,}}'=P{{{\mathcal{L}}\,}}P^\top\) for some \(P\in {{\mathrm{GL}\,}}(n)\). When \(P\in \hbox {O}(n)\), the spaces \({{{\mathcal{L}}\,}},{{{\mathcal{L}}\,}}'\) are called orthogonally congruent.‚ô¶

The sets \({{\mathrm{Jo}\,}}(m,\mathbb{S}^n)\) and \({{\mathrm{Jo}\,}}_\mathbf{1}(m,\mathbb{S}^n)\) are varieties [1] that are stable under congruence and orthogonal congruence, respectively. The goal of this section is to classify elements of \({{\mathrm{Jo}\,}}(m,\mathbb{S}^n)\) up to congruence. Every element of \({{\mathrm{Jo}\,}}(m,\mathbb{S}^n)\) is congruent to an element of \({{\mathrm{Jo}\,}}_\mathbf{1}(m,\mathbb{S}^n)\). So equivalently, we wish to classify the elements of \({{\mathrm{Jo}\,}}_\mathbf{1}(m,\mathbb{S}^n)\) up to orthogonal congruence. In this section, we list the orbits of Jordan nets and webs in low dimension.

3.1 Jordan nets

Every Jordan net is the embedding of one of the following Jordan algebras:

$$\begin{aligned} {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}},{{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^1_0,{{{\mathbb {C}}\,}}[x]/(x^3),{{\mathrm{\mathcal{J}}\,}}^2_2,{{\mathrm{\mathcal{J}}\,}}^2_1,{{\mathrm{\mathcal{J}}\,}}^2_0 \end{aligned}$$

For the first five of these algebras, we classify the orbits for general n.

Theorem 3.3

Let \(n\ge 1\) be an integer.

  1. (1)

    Every embedding of \({{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\) into \(\mathbb{S}^n\) is congruent to \(A^{(1)}_{k_1,k_2,k_3}:={{\mathrm{Diag}\,}}(x\mathbf{1}_{k_1+k_2+k_3},y\mathbf{1}_{k_2+k_3},z\mathbf{1}_{k_3})\) for some \(k_1,k_2\ge 0\) and \(k_3\ge 1\) with \(k_1+2k_2+3k_3=n\).

  2. (2)

    Every embedding of \({{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^1_0\) into \(\mathbb{S}^n\) is congruent to \(A^{(2)}_{r,k_1,k_2}:={{\mathrm{Diag}\,}}\left( x\mathbf{1}_r,\mathbf{1}_{k_2}\otimes \begin{pmatrix}z&{}y\\ y\end{pmatrix},y\mathbf{1}_{k_1}\right)\) for some \(r,k_2\ge 1\) and \(k_1\ge 0\) with \(k_1+2k_2=n-r\).

  3. (3)

    Every embedding of \({{{\mathbb {C}}\,}}[x]/(x^3)\) into \(\mathbb{S}^n\) is congruent to \(A^{(3)}_{k_1,k_2,k_3}:={{\mathrm{Diag}\,}}\left( \mathbf{1}_{k_3}\otimes \begin{pmatrix}z&{}y&{}x\\ y&{}x\\ x\end{pmatrix},\mathbf{1}_{k_2}\otimes \begin{pmatrix}y&{}x\\ x\end{pmatrix},x\mathbf{1}_{k_1}\right)\) for some \(k_1,k_2\ge 0\) and \(k_3\ge 1\) with \(k_1+2k_2+3k_3=n\).

  4. (4)

    The Jordan algebra \({{\mathrm{\mathcal{J}}\,}}^2_2\) has no embeddings into \(\mathbb{S}^n\) when n is odd. When n is even, every embedding of \({{\mathrm{\mathcal{J}}\,}}^2_2\) into \(\mathbb{S}^n\) is congruent to \(B^{(1)}_{n/2}:=\mathbb{S}^2\otimes \mathbf{1}_{n/2}\).

  5. (5)

    Every embedding of \({{\mathrm{\mathcal{J}}\,}}^2_1\) into \(\mathbb{S}^n\) is congruent to

    $$\begin{aligned}&B^{(2)}_{k,\ell _1,\ell _2}\\&\quad :={{\mathrm{Diag}\,}}\left( \begin{pmatrix}xJ_{\ell _2}&{}z{{\mathrm{Diag}\,}}(\mathbf{1}_k,\mathbf{0}_{\ell _2-k})\\ z{{\mathrm{Diag}\,}}(\mathbf{1}_k,\mathbf{0}_{\ell _2-k})&{}yJ_{\ell _2}\end{pmatrix},y \mathbf{1}_{\ell _1}\right) \end{aligned}$$

    for some \(\ell _1\ge 0\), \(\ell _2\ge 2\) and \(1\le k\le \ell _2/2\) such that \(\ell _1+2\ell _2=n\).

For embeddings of \({{\mathrm{\mathcal{J}}\,}}^2_0\) we classify the orbits of embeddings into \(\mathbb{S}^n\) for \(n\le 7\).

Theorem 3.4

For \(n\in \{1,2,3\}\), the Jordan algebra \({{\mathrm{\mathcal{J}}\,}}^2_0\) has no embeddings into \(\mathbb{S}^n\). For \(4\le n\le 7\), every embedding of \({{\mathrm{\mathcal{J}}\,}}^2_0\) into \(\mathbb{S}^n\) is congruent \(C_{n,i}={{{\mathbb {C}}\,}}J_n+{{\mathrm{Diag}\,}}(\mathcal{P}_{\lfloor n/2\rfloor ,i},\mathbf{0}_{\lceil n/2\rceil })\) for some i where

$$\begin{aligned}&\mathcal{P}_{2,1}:=\begin{pmatrix}x\\ {} &{}y\end{pmatrix},&&\mathcal{P}_{2,2}:=\begin{pmatrix}y&{}x\\ x\end{pmatrix},\\&\mathcal{P}_{3,1}:=\begin{pmatrix}x\\ {} &{}y\\ &{}&{}x+y\end{pmatrix},&&\mathcal{P}_{3,2}:=\begin{pmatrix}x&{}y\\ y\\ &{}&{}x\end{pmatrix},\\&\mathcal{P}_{3,3}:=\begin{pmatrix}&{}y&{}x\\ y&{}x\\ x\end{pmatrix},&&\mathcal{P}_{3,4}:=\begin{pmatrix}x\\ {} &{}x\\ &{}&{}y\end{pmatrix},\\&\mathcal{P}_{3,5}:=\begin{pmatrix}y&{}x\\ x\\ &{}&{}x\end{pmatrix},&&\mathcal{P}_{3,6}:=\begin{pmatrix}&{}x&{}y\\ x\\ y\end{pmatrix},\\&\mathcal{P}_{3,7}:=\begin{pmatrix}x\\ {} &{}y\\ &{}&{}0\end{pmatrix},&&\mathcal{P}_{3,8}:=\begin{pmatrix}y&{}x\\ x\\ &{}&{}0\end{pmatrix}. \end{aligned}$$

Proof of Theorem 3.3

  1. (1)

    Every embedding of \({{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\) is congruent to \({{\mathrm{Diag}\,}}({{{\mathcal{L}}\,}}_1,{{{\mathcal{L}}\,}}_2,{{{\mathcal{L}}\,}}_3)\) for some embeddings \({{{\mathcal{L}}\,}}_1,{{{\mathcal{L}}\,}}_2,{{{\mathcal{L}}\,}}_3\) of \({{{\mathbb {C}}\,}}\) by Proposition A. 5. By Proposition A. 6, we may assume that \({{{\mathcal{L}}\,}}_i={{{\mathbb {C}}\,}}\mathbf{1}_{n_i}\) for some \(n_i\ge 1\). After reordening, we can write \(n_1=k_1+k_2+k_3\), \(n_2=k_2+k_3\) and \(n_3=k_3\) for some \(k_1,k_2\ge 0\) and \(k_3\ge 1\) such that \(k_1+2k_2+3k_3=n\).

  2. (2)

    Every embedding of \({{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^1_0\) is congruent to \({{\mathrm{Diag}\,}}({{{\mathcal{L}}\,}}_1,{{{\mathcal{L}}\,}}_2)\) for some embeddings \({{{\mathcal{L}}\,}}_1\) of \({{{\mathbb {C}}\,}}\) and \({{{\mathcal{L}}\,}}_2\) of \({{\mathrm{\mathcal{J}}\,}}^1_0\) by Proposition A. 5. By Proposition A. 6, we may assume that \({{{\mathcal{L}}\,}}_1={{{\mathbb {C}}\,}}\mathbf{1}_r\) for some \(r\ge 1\). By Proposition A. 7, we may assume that

    $$\begin{aligned} {{{\mathcal{L}}\,}}_2={{\mathrm{Diag}\,}}\left( \mathbf{1}_{k_2}\otimes \begin{pmatrix}z&{}y\\ y\end{pmatrix},y\mathbf{1}_{k_1}\right) \end{aligned}$$

    for some \(k_1\ge 0\) and \(k_2\ge 1\) with \(k_1+2k_2=n-r\).

  3. (3)

    This is Proposition A. 7.

  4. (4)

    This is Proposition A. 18.

  5. (5)

    This is Proposition A. 17.

\(\square\)

Proof of Theorem 3.4

  1. (1)

    A pencil in \(\mathbb{S}^n\) always contains a matrix of rank \(\ge 2\). On the other hand, all matrices in \(\mathbb{S}^n\) whose squares are zero have rank \(\le n/2\). So square-zero pencils cannot exist in \(\mathbb{S}^n\) when \(n<4\).

  2. (2)

    This follows from Propositions A. 11 and A. 14.

  3. (3)

    This follows from Propositions A. 11 and A. 14.

  4. (4)

    This follows from Propositions A. 20 and A. 15.

  5. (5)

    This follows from Propositions A. 20 and A. 15.

\(\square\)

3.2 Jordan webs

Every Jordan web is the embedding of one of the following Jordan algebras:

$$\begin{aligned} &{{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}},{{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^1_0,{{\mathrm{\mathcal{J}}\,}}^1_0\times {{\mathrm{\mathcal{J}}\,}}^1_0,{{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}[x]/(x^3),{{{\mathbb {C}}\,}}[x]/(x^4),\\&{{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^2_2,{{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^2_1,{{\mathrm{\mathcal{J}}\,}}^3_3,{{\mathrm{\mathcal{J}}\,}}^3_2, {{\mathrm{\mathcal{J}}\,}}^3_1,{{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^2_0,{{\mathrm{\mathcal{E}}\,}}_1,{{\mathrm{\mathcal{E}}\,}}_2,{{\mathrm{\mathcal{E}}\,}}_3,{{\mathrm{\mathcal{E}}\,}}_4,{{\mathrm{\mathcal{J}}\,}}^3_0. \end{aligned}$$

For the algebras on the first line, we classify the orbits for general n.

Theorem 3.5

Let \(n\ge 1\) be an integer.

  1. (1)

    Every embedding of \({{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\) into \(\mathbb{S}^n\) is congruent to \(A^{(1)}_{k_1,k_2,k_3,k_4}:={{\mathrm{Diag}\,}}(x\mathbf{1}_{k_1+k_2+k_3+k_4},y\mathbf{1}_{k_2+k_3+k_4},z\mathbf{1}_{k_3+k_4},w\mathbf{1}_{k_4})\) for some \(k_1,k_2,k_3\ge 0\) and \(k_4\ge 1\) with \(k_1+2k_2+3k_3+4k_4=n\).

  2. (2)

    Every embedding of \({{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^1_0\) into \(\mathbb{S}^n\) is congruent to \(A^{(2)}_{k_1,k_2,\ell _1,\ell _2}:={{\mathrm{Diag}\,}}\left( x\mathbf{1}_{k_1+k_2},y\mathbf{1}_{k_2},\mathbf{1}_{\ell _2}\otimes \begin{pmatrix}w&{}z\\ z\end{pmatrix},z\mathbf{1}_{\ell _1}\right)\) for some \(k_1,\ell _1\ge 0\) and \(k_2,\ell _2\ge 1\) with \(k_1+2k_2+\ell _1+2\ell _2=n\).

  3. (3)

    Every embedding of \({{\mathrm{\mathcal{J}}\,}}^1_0\times {{\mathrm{\mathcal{J}}\,}}^1_0\) into \(\mathbb{S}^n\) is congruent to \(A^{(3)}_{k_1,k_2,\ell _1,\ell _2}:={{\mathrm{Diag}\,}}\left( \mathbf{1}_{k_2}\otimes \begin{pmatrix}y&{}x\\ x\end{pmatrix},x\mathbf{1}_{k_1},\mathbf{1}_{\ell _2}\otimes \begin{pmatrix}w&{}z\\ z\end{pmatrix},z\mathbf{1}_{\ell _1}\right)\) for some \(k_1,\ell _1\ge 0\) and \(k_2,\ell _2\ge 1\) with \(k_1+2k_2+\ell _1+2\ell _2=n\), where \(A^{(3)}_{k_1,k_2,\ell _1,\ell _2}\) and \(A^{(3)}_{\ell _1,\ell _2,k_1,k_2}\) are congruent.

  4. (4)

    Every embedding of \({{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}[x]/(x^3)\) into \(\mathbb{S}^n\) is congruent to \(A^{(4)}_{r,k_1,k_2,k_3}:={{\mathrm{Diag}\,}}\left( w\mathbf{1}_r,\mathbf{1}_{k_3}\otimes \begin{pmatrix}z&{}y&{}x\\ y&{}x\\ x\end{pmatrix},\mathbf{1}_{k_2}\otimes \begin{pmatrix}y&{}x\\ x\end{pmatrix},x\mathbf{1}_{k_1}\right)\) for some \(k_1,k_2\ge 0\) and \(r,k_3\ge 1\) with \(k_1+2k_2+3k_3=n-r\).

  5. (5)

    Every embedding of \({{{\mathbb {C}}\,}}[x]/(x^4)\) into \(\mathbb{S}^n\) is congruent to \(A^{(5)}_{k_1,k_2,k_3,k_4}:={{\mathrm{Diag}\,}}\left( \mathbf{1}_{k_4}\otimes \begin{pmatrix}w&{}z&{}y&{}x\\ z&{}y&{}x\\ y&{}x\\ x\end{pmatrix},\mathbf{1}_{k_3}\otimes \begin{pmatrix}z&{}y&{}x\\ y&{}x\\ x\end{pmatrix},\mathbf{1}_{k_2}\otimes \begin{pmatrix}y&{}x\\ x\end{pmatrix},x\mathbf{1}_{k_1}\right)\) for some \(k_1,k_2,k_3\ge 0\) and \(k_4\ge 1\) with \(k_1+2k_2+3k_3+4k_4=n\).

  6. (6)

    Every embedding of \({{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^2_2\) into \(\mathbb{S}^n\) is congruent to \(B^{(1)}_{k_1,k_2}:={{\mathrm{Diag}\,}}\left( w\mathbf{1}_{k_1},\begin{pmatrix}x&{}y\\ y&{}z\end{pmatrix}\otimes \mathbf{1}_{k_2}\right)\) for some \(k_1,k_2\ge 1\) with \(k_1+2k_2=n\).

  7. (7)

    Every embedding of \({{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^2_1\) into \(\mathbb{S}^n\) is congruent to \(B^{(2)}_{r,k,\ell _1,\ell _2}:={{\mathrm{Diag}\,}}\left( w\mathbf{1}_r,\begin{pmatrix}xJ_{\ell _2}&{}z{{\mathrm{Diag}\,}}(\mathbf{1}_k,\mathbf{0}_{\ell _2-k})\\ z{{\mathrm{Diag}\,}}(\mathbf{1}_k,\mathbf{0}_{\ell _2-k})&{}yJ_{\ell _2}\end{pmatrix},y \mathbf{1}_{\ell _1}\right)\) for some \(\ell _1\ge 0\), \(r\ge 1\), \(\ell _2\ge 2\) and \(1\le k\le \ell _2/2\) such that \(\ell _1+2\ell _2=n-r\).

  8. (8)

    The Jordan algebra \({{\mathrm{\mathcal{J}}\,}}^3_3\) has no embeddings into \(\mathbb{S}^n\) when \(4\not \mid n\). When \(4\mid n\), every embedding of \({{\mathrm{\mathcal{J}}\,}}_3^3\) into \(\mathbb{S}^n\) is congruent to \(C^{(1)}_{n/4}:=\begin{pmatrix}x\mathbf{1}_{n/2}&{}z\mathbf{1}_{n/2}+w\mathbf{1}_{n/4}\otimes \begin{pmatrix}&{}1\\ -1\end{pmatrix}\\ z\mathbf{1}_{n/2}+w\mathbf{1}_{n/4}\otimes \begin{pmatrix}&{}-1\\ 1\end{pmatrix}&y\mathbf{1}_{n/2}\end{pmatrix}.\)

  9. (9)

    The Jordan algebra \({{\mathrm{\mathcal{J}}\,}}^3_2\) has no embeddings into \(\mathbb{S}^n\) when n is odd. When n is even, every embedding of \({{\mathrm{\mathcal{J}}\,}}_2^3\) into \(\mathbb{S}^n\) is congruent to \(C^{(2)}_{n/2,k}:=\begin{pmatrix}x\mathbf{J}_{n/2}&{}\!z\mathbf{J}_{n/2}+w{{\mathrm{Diag}\,}}\!\left( \!\mathbf{1}_k\!\otimes \!\begin{pmatrix}&{}1\\ -1\!\!\!\!\!\end{pmatrix}\!,\mathbf{0}_{n/2-2k}\!\right) \!\\ \!z\mathbf{J}_{n/2}+w{{\mathrm{Diag}\,}}\!\left( \!\mathbf{1}_k\!\otimes \!\begin{pmatrix}&{}-1\\ 1\!\!\!\!\!\end{pmatrix}\!\!,\mathbf{0}_{n/2-2k}\!\right) \!&y\mathbf{J}_{n/2}\end{pmatrix}\) for some integer \(1\le k\le n/8\).

For embeddings of \({{\mathrm{\mathcal{J}}\,}}^3_1,{{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^2_0,{{\mathrm{\mathcal{E}}\,}}_1,{{\mathrm{\mathcal{E}}\,}}_2,{{\mathrm{\mathcal{E}}\,}}_3,{{\mathrm{\mathcal{E}}\,}}_4,{{\mathrm{\mathcal{J}}\,}}^3_0\) we classify the orbits of embeddings into \(\mathbb{S}^n\) for \(n\le 5\).

Proposition 3.6

The Jordan algebras \({{\mathrm{\mathcal{J}}\,}}^3_1,{{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^2_0,{{\mathrm{\mathcal{E}}\,}}_2,{{\mathrm{\mathcal{E}}\,}}_3,{{\mathrm{\mathcal{E}}\,}}_4,{{\mathrm{\mathcal{J}}\,}}^3_0\) have no embeddings into \(\mathbb{S}^3\). Every embedding of \({{\mathrm{\mathcal{E}}\,}}_1\) into \(\mathbb{S}^3\) is congruent to \(E^{(1)}_3:={{\mathrm{\mathcal{E}}\,}}_1\).

Theorem 3.7

  1. (1)

    The Jordan algebra \({{\mathrm{\mathcal{J}}\,}}^3_1\) has no embedding into \(\mathbb{S}^4\).

  2. (2)

    The Jordan algebra \({{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^2_0\) has no embedding into \(\mathbb{S}^4\).

  3. (3)

    Every embedding of \({{\mathrm{\mathcal{E}}\,}}_1\) into \(\mathbb{S}^4\) is congruent to one of

    $$\begin{aligned} E^{(1)}_{4,1}:=\begin{pmatrix}v&{}w&{}x\\ w&{}y\\ x\\ &{}&{}&{}x\end{pmatrix},E^{(1)}_{4,2}:=\begin{pmatrix}v&{}w&{}x\\ w&{}y\\ x\\ &{}&{}&{}y\end{pmatrix}. \end{aligned}$$
  4. (4)

    Every embedding of \({{\mathrm{\mathcal{E}}\,}}_2\) into \(\mathbb{S}^4\) is congruent to

    $$\begin{aligned} E^{(2)}_4:=\begin{pmatrix}v&{}x&{}w\\ x\\ w&{}&{}&{}y\\ &{}&{}y\end{pmatrix}. \end{aligned}$$
  5. (5)

    Every embedding of \({{\mathrm{\mathcal{E}}\,}}_3\) into \(\mathbb{S}^4\) is congruent to

    $$\begin{aligned} E^{(3)}_4:=\begin{pmatrix}y&{}x&{}u&{}z\\ x&{}u\\ u\\ z&{}&{}&{}u\end{pmatrix}. \end{aligned}$$
  6. (6)

    The Jordan algebra \({{\mathrm{\mathcal{E}}\,}}_4\) has no embeddings into \(\mathbb{S}^4\).

  7. (7)

    Every embedding of \({{\mathrm{\mathcal{J}}\,}}^3_0\) into \(\mathbb{S}^4\) is congruent to

    $$\begin{aligned} F_4 := \begin{pmatrix} x&{}y&{}&{}u\\ y&{}z&{}u\\ {} &{}u\\ u \end{pmatrix}. \end{aligned}$$

Theorem 3.8

  1. (1)

    The Jordan algebra \({{\mathrm{\mathcal{J}}\,}}^3_1\) has no embedding into \(\mathbb{S}^5\).

  2. (2)

    Every embedding of \({{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^2_0\) into \(\mathbb{S}^5\) is congruent to one of

    $$\begin{aligned} D_{5,1}:=\begin{pmatrix}x\\ {} &{}v&{}&{}&{}y\\ &{}&{}w&{}y\\ &{}&{}y\\ {} &{} y\end{pmatrix},~ D_{5,2}:=\begin{pmatrix}x\\ {} &{}v&{}w&{}&{}y\\ {} &{}w&{}&{}y\\ &{}&{}y\\ {} &{}y\end{pmatrix}. \end{aligned}$$
  3. (3)

    Every embedding of \({{\mathrm{\mathcal{E}}\,}}_1\) into \(\mathbb{S}^5\) is congruent to one of

    $$\begin{aligned} E^{(1)}_{5,1}:=\begin{pmatrix}v&{}w&{}x\\ w&{}y\\ x\\ &{}&{}&{}x\\ &{}&{}&{}&{}x\end{pmatrix},E^{(1)}_{5,2}:=\begin{pmatrix}v&{}w&{}x\\ w&{}y\\ x\\ &{}&{}&{}x\\ &{}&{}&{}&{}y\end{pmatrix},E^{(1)}_{5,3}:=\begin{pmatrix}v&{}w&{}x\\ w&{}y\\ x\\ &{}&{}&{}y\\ &{}&{}&{}&{}y\end{pmatrix}. \end{aligned}$$
  4. (4)

    Every embedding of \({{\mathrm{\mathcal{E}}\,}}_2\) into \(\mathbb{S}^5\) is congruent to one of

    $$\begin{aligned} E^{(2)}_{5,1}:=\begin{pmatrix}v&{}x&{}w\\ x\\ w&{}&{}&{}y\\ &{}&{}y\\ &{}&{}&{}&{}x\end{pmatrix},E^{(2)}_{5,2}:=\begin{pmatrix}v&{}x&{}w\\ x\\ w&{}&{}&{}y\\ &{}&{}y\\ &{}&{}&{}&{}y\end{pmatrix}. \end{aligned}$$
  5. (5)

    Every embedding of \({{\mathrm{\mathcal{E}}\,}}_3\) into \(\mathbb{S}^5\) is congruent to one of

    $$\begin{aligned} E^{(3)}_{5,1}:=\begin{pmatrix}y&{}x&{}u&{}z\\ x&{}u\\ u\\ z&{}&{}&{}u\\ &{}&{}&{}&{}u\end{pmatrix},E^{(3)}_{5,2}:=\begin{pmatrix}y&{}x&{}u&{}&{}z\\ x&{}u\\ u\\ &{}&{}&{}x&{}u\\ z&{}&{}&{}u\end{pmatrix}. \end{aligned}$$
  6. (6)

    Every embedding of \({{\mathrm{\mathcal{E}}\,}}_4\) into \(\mathbb{S}^5\) is congruent to one of

    $$\begin{aligned} E^{(4)}_{5,1}:=\begin{pmatrix}y&{}x&{}u\\ x&{}u\\ u\\ &{}&{}&{}z&{}u\\ &{}&{}&{}u\end{pmatrix},E^{(4)}_{5,2}:=\begin{pmatrix}y&{}x&{}u&{}z\\ x&{}u\\ u\\ z&{}&{}&{}&{}u\\ &{}&{}&{}u\end{pmatrix}. \end{aligned}$$
  7. (7)

    Every embedding of \({{\mathrm{\mathcal{J}}\,}}^3_0\) into \(\mathbb{S}^5\) is congruent to

    $$\begin{aligned} F_5:=\begin{pmatrix} x&{}y&{}&{}&{}u\\ y&{}z&{}&{}u\\ &{}&{}u\\ {} &{}u\\ u \end{pmatrix}. \end{aligned}$$

Proof of Theorem 3.5

  1. (1)

    Every embedding of \({{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\) is congruent to \({{\mathrm{Diag}\,}}({{{\mathcal{L}}\,}}_1,{{{\mathcal{L}}\,}}_2,{{{\mathcal{L}}\,}}_3,{{{\mathcal{L}}\,}}_4)\) for some embeddings \({{{\mathcal{L}}\,}}_1,{{{\mathcal{L}}\,}}_2,{{{\mathcal{L}}\,}}_3,{{{\mathcal{L}}\,}}_4\) of \({{{\mathbb {C}}\,}}\) by Proposition A. 5. By Proposition A. 6, we may assume that \({{{\mathcal{L}}\,}}_i={{{\mathbb {C}}\,}}\mathbf{1}_{n_i}\) for some \(n_i\ge 1\). After reordening, we can write \(n_1=k_1+k_2+k_3+k_4\), \(n_2=k_2+k_3+k_4\), \(n_3=k_3+k_4\) and \(n_4=k_4\) for some \(k_1,k_2,k_3\ge 0\) and \(k_4\ge 1\) such that \(k_1+2k_2+3k_3+4k_4=n\).

  2. (2)

    This follows from Propositions A. 5, A. 6 and A. 7.

  3. (3)

    This follows from Propositions A. 5 and A. 7.

  4. (4)

    This follows from Propositions A. 5, A. 6 and A. 7.

  5. (5)

    This is Proposition A. 7.

  6. (6)

    This follows from Propositions A. 5, A. 6 and A. 18.

  7. (7)

    This follows from Propositions A. 5, A. 6 and A. 17.

  8. (8)

    This is Proposition A. 23.

  9. (9)

    This is Proposition A. 22.

\(\square\)

Proof of Proposition 3.6

By [1, Proposition 4.8], we know that inside \(\mathbb{S}^3\) there are two orbits of Jordan space, consisting of embeddings of the Jordan algebras \({{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^2_2\) and \({{\mathrm{\mathcal{E}}\,}}_1\). The proposition follows. \(\square\)

Proof of Theorem 3.7

  1. (1)

    This is Proposition A. 21.

  2. (2)

    This follows by Proposition A. 5 since \({{\mathrm{\mathcal{J}}\,}}^2_0\) has no embeddings into \(\mathbb{S}^n\) for \(n\le 3\).

  3. (3)

    This is Proposition A. 24.

  4. (4)

    This is Proposition A. 25.

  5. (5)

    This is Proposition A. 26.

  6. (6)

    This is Proposition A. 27.

  7. (7)

    This follows from Propositions A. 19 and A. 20.

\(\square\)

Proof of Theorem 3.8

  1. (1)

    This is Proposition A. 21.

  2. (2)

    This follows from Propositions A. 5, A. 6, A. 9 and A. 14.

  3. (3)

    This is Proposition A. 24.

  4. (4)

    This is Proposition A. 25.

  5. (5)

    This is Proposition A. 26.

  6. (6)

    This is Proposition A. 27.

  7. (7)

    This follows from Propositions A. 19 and A. 20.

\(\square\)

4 Degenerations between Jordan nets

Let \({{{\mathcal{L}}\,}},{{{\mathcal{L}}\,}}'\subseteq \mathbb{S}^n\) be Jordan spaces of the same dimension m with bases \(X_1,\ldots ,X_m\) and \(X'_1,\ldots ,X'_m\).

Definition 4.1

We say that \({{{\mathcal{L}}\,}}\) degenerates to \({{{\mathcal{L}}\,}}'\), denoted as \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\), when there exist matrices \(P\in {{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}}[t^{\pm 1}])\) and \(Q\in {{\mathrm{GL}\,}}_m({{{\mathbb {C}}\,}}[t^{\pm 1}])\) such that every entry of

$$\begin{aligned} (Y_1,\ldots ,Y_m):= (PX_1P^\top ,\ldots ,PX_mP^\top )Q \end{aligned}$$

is a matrix with coefficients in \({{{\mathbb {C}}\,}}[t]\) and \(X'_i=\lim _{t\rightarrow 0}Y_i\) for all \(i\in \{1,\ldots ,m\}\). Note that the existence of such matrices P, Q does not depend on the choice of the bases of \({{{\mathcal{L}}\,}},{{{\mathcal{L}}\,}}'\). We say that \({{{\mathcal{L}}\,}}\) degenerates to \({{{\mathcal{L}}\,}}'\) (topologically) when \(\overline{\{(Y_1,\ldots ,Y_m)\mid {{\mathrm{span}\,}}(Y_1,\ldots ,Y_m)\in {{\mathrm{GL}\,}}_n\cdot {{{\mathcal{L}}\,}}\}}\) contains \((X_1',\ldots ,X_m')\).♦

For a proof that these definitions are equivalent, see the proof of [3, Theorem 20.24].

Remark 4.2

Note that \({{{\mathbb {C}}\,}}[t^{\pm 1}]\) is a subalgebra of \({{{\mathbb {C}}\,}}((t))\) and \({{{\mathbb {C}}\,}}[t]\) is a subalgebra of \({{{\mathbb {C}}\,}}[[t]]\). So if \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\), then there exist matrices \(P\in {{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}}((t)))\) and \(Q\in {{\mathrm{GL}\,}}_m({{{\mathbb {C}}\,}}((t)))\) such that every entry of

$$\begin{aligned} (Y_1,\ldots ,Y_m):= (PX_1P^\top ,\ldots ,PX_mP^\top )Q \end{aligned}$$

is a matrix with coefficients in \({{{\mathbb {C}}\,}}[[t]]\) and \(X'_i=\lim _{t\rightarrow 0}Y_i\) for all \(i\in \{1,\ldots ,m\}\). The converse also holds: let \(\ell \ge 0\) be an integer such that the coefficients of P, Q are contained in \(t^{-\ell }{{{\mathbb {C}}\,}}[[t]]\) and write \((P,Q)=\sum _{k=-\ell }^\infty t^k(P_k,Q_k)\) with \((P_k,Q_k)\in {{{\mathbb {C}}\,}}^{n\times n}\times {{{\mathbb {C}}\,}}^{m\times m}\). Then \(\lim _{t\rightarrow 0}\det (P)\), \(\lim _{t\rightarrow 0}\det (Q)\) and \(\lim _{t\rightarrow 0}Y_i\) do not depend on the \(P_k\)’s, \(Q_k\)’s with \(k>\ell \max (3,n,m)\). Hence, we are free to set these to zero and obtain matrices \(P\in {{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}}[t^{\pm 1}])\) and \(Q\in {{\mathrm{GL}\,}}_m({{{\mathbb {C}}\,}}[t^{\pm 1}])\) showing that \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\).♣

In this section, we give the diagrams of congruence orbits of Jordan nets in \(\mathbb{S}^n\) and their degenerations for \(n\le 6\). See Sect. 8 for the proofs of these results.

For \(n=2\), the whole space \(\mathbb{S}^2\) is the only Jordan net.

Fig. 1
figure 1

Jordan nets in \(\mathbb{S}^2\) and their degenerations

For \(n=3\), we have 3 orbits which form a chain. For \(n=4\), the degenerations where classified in [1]. See Figs. 1, 2 and 3.

Fig. 2
figure 2

Jordan nets in \(\mathbb{S}^3\) and their degenerations

Fig. 3
figure 3

Jordan nets in \(\mathbb{S}^4\) and their degenerations

The first new case is that of \(n=5\). We have the following result.

Fig. 4
figure 4

Jordan nets in \(\mathbb{S}^5\) and their degenerations

Theorem 4.3

The diagram in Fig. 4 describes all degenerations of Jordan nets in \(\mathbb{S}^5\).

Next is the case of \(n=6\). We have the following conjecture and weaker statement..

Conjecture 4.4

We have \(B^{(1)}_3\not \rightarrow C_{6,6}\).

Proposition 4.5

We have \(B^{(2)}_{1,0,3}\not \rightarrow C_{6,6}\).

Assuming the conjecture, the diagram in Fig. 5 is already complete.

Theorem 4.6

Apart from possibly the dotted line, the diagram in Fig. 5 describes all degenerations of Jordan nets in \(\mathbb{S}^6\).

Fig. 5
figure 5

Jordan nets in \(\mathbb{S}^6\) and their degenerations. The dotted lines indicate that we do not know whether these degenerations exist, but we believe that they do not

5 Numerical results for bigger n and Jordan webs

This section is devoted to finding degenerations between embedded Jordan algebras in the sense of Definition 4.1. A basis \(X_1,\ldots ,X_m\) of an embedded Jordan algebra \({{{\mathcal{L}}\,}}\subseteq \mathbb{S}^n\) gives the \(n\times n\times m\) tensor

$$\begin{aligned} {\mathbf {X}}=[X_1|\dots |X_m] \end{aligned}$$

and, for \(P_1,P_2\in {{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}})\) and \(Q\in {{\mathrm{GL}\,}}_m({{{\mathbb {C}}\,}})\), we write \([\![{\mathbf {X}};P_1,P_2,Q]\!]\) for the result of acting on the rows, columns and layers of \({\mathbf {X}}\) by \(P_1,P_2,Q\), respectively. Definition 4.1 is then equivalent to

$$\begin{aligned} \lim _{t \rightarrow 0} [\![{\mathbf {X}};P(t),P(t),Q(t)]\!]={\mathbf {Y}} \end{aligned}$$

for some \(P\in {{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}}[t^{\pm 1}])\) and \(Q\in {{\mathrm{GL}\,}}_m({{{\mathbb {C}}\,}}[t^{\pm 1}])\) and a corresponding basis tensor \({\mathbf {Y}}\) for \({{{\mathcal{L}}\,}}'\). Given a basis tensor \({\mathbf {X}}\) of \({{{\mathcal{L}}\,}}\), the basis tensors of \({{{\mathcal{L}}\,}}'\) in the orbit of \({{{\mathcal{L}}\,}}\) are \([\![{\mathbf {X}};P,P,Q]\!]\) for \(P\in {{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}})\) and \(Q\in {{\mathrm{GL}\,}}_m({{{\mathbb {C}}\,}})\), and if \({\mathbf {Y}}\) is a basis tensor of a degeneration of \({{{\mathcal{L}}\,}}\), then \({\mathbf {Y}}\) lies in the Zariski closure of the polynomial map

$$\begin{aligned} {{{\mathbb {C}}\,}}^{n\times n}\times {{{\mathbb {C}}\,}}^{m\times m}\rightarrow & {} {{{\mathbb {C}}\,}}^{n\times n\times m}\\ (P,Q)\mapsto & {} [\![{\mathbf {X}};P,P,Q]\!]. \end{aligned}$$

On the other hand, if \({\mathbf {Y}}\) does not lie in the Zariski closure, then the corresponding embedded Jordan algebra is no degeneration of \({{{\mathcal{L}}\,}}\) since the closures in the Euclidean and Zariski topologies coincide. A definite answer can be given by eliminating the variables (P, Q) from the ideal \(\mathcal{I}\subset {{{\mathbb {C}}\,}}[P,Q,Z]\) generated by the equations \(Z- [\![{\mathbf {X}};P,P,Q]\!]\): If \({\mathbf {Y}}\in \mathcal{V}(\mathcal{I}\cap {{{\mathbb {C}}\,}}[Z])\), then \({\mathbf {Y}}\) is a basis tensor of a degeneration of \({{{\mathcal{L}}\,}}\). This is, however, only feasible for very small n and m.

Instead, we use gradient descent algorithms to find the distance

$$\begin{aligned} \inf _{\begin{array}{c} P\in {{{\mathbb {C}}\,}}^{n\times n}\\ Q\in {{{\mathbb {C}}\,}}^{m\times m} \end{array}} f(P,Q)=\inf _{\begin{array}{c} P\in {{{\mathbb {C}}\,}}^{n\times n}\\ Q\in {{{\mathbb {C}}\,}}^{m\times m} \end{array}} \Vert {\mathbf {Y}}-[\![{\mathbf {X}};P,P,Q]\!]\Vert ^2 \end{aligned}$$

of the orbit

$$\begin{aligned} \{ [\![{\mathbf {X}};P,P,Q]\!]\mid P\in {{{\mathbb {C}}\,}}^{n\times n}, \; Q\in {{{\mathbb {C}}\,}}^{m\times m}\} \end{aligned}$$

of \({\mathbf {X}}\) to \({\mathbf {Y}}\). This way can find a sequence \((P_i,Q_i)\) such that \(\lim _{i\rightarrow \infty }f(P_i,Q_i)= 0\) when \({\mathbf {Y}}\) is a basis of a degeneration of \({{{\mathcal{L}}\,}}\) with basis tensor \({\mathbf {X}}\). We will say \({\mathbf {X}}\) degenerates to \({\mathbf {Y}}\) if \(\inf _{P,Q} f(P,Q)=0\). The function f is a real smooth function in \(2(n^2+m^2)\) real variables. We can either use gradient descent methods directly for f or we can use gradient methods for

$$\begin{aligned} \inf _{P\in {{{\mathbb {C}}\,}}^{n\times n}} F(P)=\inf _{P\in {{{\mathbb {C}}\,}}^{n\times n}}\left( \inf _{Q\in {{{\mathbb {C}}\,}}^{n\times n}}f(P,Q)\right) =\inf _{P\in {{{\mathbb {C}}\,}}^{n\times n}}\left( \min _{Q\in {{{\mathbb {C}}\,}}^{n\times n}}f(P,Q)\right) , \end{aligned}$$

where the infimum is attained since f is quadratic and convex in Q. We observed that using gradient methods for F instead of f was more effective. This might be a result of having less variables to minimize over. The value of F at P can be computed as the solution of a linear equation. If the minimizer Q(P) is unique in a neighborhood of P, we can compute the gradient of F at P via the chain rule using \(F(P)=f(P,Q(P))\) and we get \(\nabla F(P)=\nabla _P f(P,Q)\) where Q is the minimizer in the definition of F. If F is not smooth at P, i.e., if the minimizer Q is not unique, then its generalized gradient in the sense of [4, Definition 1.1] is given by the convex hull of

$$\begin{aligned} \{\nabla _P f(P,Q)\mid \hbox {} Q \hbox {is minimizer of}\; f(P,Q)\}. \end{aligned}$$

In practice we used the BFGS method to minimize F in order to utilize second-order information. A major obstacle is that both f and F are highly non-convex. Therefore, there can be many local minima. Hence, to find degenerations we therefore used gradient descents with 50 randomly generated starting guesses \(P_0\in {{{\mathbb {C}}\,}}^{n\times n}\) to have higher odds of finding the global minimum of F.

Another obstacle is that for degenerations only a minimizing sequence exists, i.e., the infimum value \(F=0\) is not attained. It is therefore not entirely obvious at which value of F we have found a degeneration. Note that infimum value for nondegenerations is not only dependent on the choice of Jordan algebra in the orbit but even on the choice of basis tensor \({\mathbf {Y}}\) of \({{{\mathcal{L}}\,}}'\). To tackle one of these issues, we always use an orthonormal basis. We also use the following idea: let \({\mathbf {X}},{\mathbf {Y}},{\mathbf {Z}}\) be basis tensors and take

$$\begin{aligned} &f(P,Q)= {} \Vert {\mathbf {Y}}-[\![{\mathbf {X}};P,P,Q]\!]\Vert ^2, \\& g(P,Q)= {} \Vert {\mathbf {Z}}-[\![{\mathbf {Y}};P,P,Q]\!]\Vert ^2, \\& h(P,Q)= \Vert {\mathbf {Z}}-[\![{\mathbf {X}};P,P,Q]\!]\Vert ^2. \end{aligned}$$

If \(\inf _{P,Q} f(P,Q)=0\), then

$$\begin{aligned} \inf _{P,Q} h(P,Q)&=\inf _{P_1,P_2,Q_1,Q_2} \Vert {\mathbf {Z}}-[\![{\mathbf {Y}};P_1,P_1,Q_1]\!]+[\![{\mathbf {Y}};P_1,P_1,Q_2]\!]-[\![{\mathbf {X}};\, P_2P_1,P_2P_1,Q_2Q_1]\!]\Vert ^2\\&\le \inf _{P_1,P_2,Q_1,Q_2} \left( \Vert {\mathbf {Z}}-[\![{\mathbf {Y}};\, P_1,P_1,Q_1]\!]\Vert +\Vert [\![{\mathbf {Y}};P_1,P_1,Q_1]\!]-[\![{\mathbf {X}};P_2P_1,P_2P_1,Q_2Q_1]\!]\Vert \right) ^2\\&=\inf _{P,Q} g(P,Q) \end{aligned}$$

where the last equality holds since \([\![{\mathbf {Y}};P_1,P_1,Q_1]\!]\) is also a basis tensor of a degeneration of \({\mathbf {X}}\) if \({\mathbf {Y}}\) is one. Now suppose that we know that \({\mathbf {X}}\) degenerates to \({\mathbf {Y}}\). Then the distance of the orbit of \({\mathbf {X}}\) to \({\mathbf {Z}}\) is not larger than the distance of the orbit of \({\mathbf {Y}}\) to \({\mathbf {Z}}\). So if \(h(P_1,Q_1)\ge g(P_2,Q_2)+\varepsilon\) for some \(\varepsilon >0\), we know that \(h(P_1,Q_1)\ge \inf _{P,Q} h(P,Q)+\varepsilon\) is not close to the minimum.

As a first experiment we confirmed Theorem 4.3 numerically. For this, we used the orthonormal bases given by the Jordan algebras described in Theorems 3.3 and 3.4. The results are summarized in Table 1 and confirm Theorem 4.3.

Table 1 Squared distances found between orbits of the Jordan nets in \(\mathbb{S}^5\) via gradient descent

As a second experiment we confirmed all degenerations for Jordan nets in \(\mathbb{S}^6\) in Fig. 5. For Conjecture 4.4, we have found further evidence. The smallest value of F for basis tensors \({\mathbf {X}}\) and \({\mathbf {Y}}\) of the Jordan algebras \(B^{(1)}_3\) and \(C_{6,6}\) was 1.0 which suggests, that there is indeed no degeneration.

For Jordan nets in \(\mathbb{S}^7\) we found the Hasse diagram in Fig. 6.

Fig. 6
figure 6

Jordan nets in \(\mathbb{S}^7\) and their degenerations obtained numerically

Fig. 7
figure 7

Jordan webs in \(\mathbb{S}^5\) and their degenerations obtained numerically

For Jordan webs in \(\mathbb{S}^4\) and \(\mathbb{S}^5\) we found the diagrams in Figs.  7 and 8.

Fig. 8
figure 8

Jordan webs in \(\mathbb{S}^4\) and their degenerations obtained numerically

6 Obstructions to degenerations between Jordan spaces

The goal of this section is to make a list of obstructions to the existence of degenerations \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\).

6.1 The abstract obstruction

Suppose that \({{{\mathcal{L}}\,}},{{{\mathcal{L}}\,}}'\) are embeddings of Jordan algebras \({{\mathrm{\mathcal{A}}\,}},{{\mathrm{\mathcal{A}}\,}}'\).

Proposition 6.1

If \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\), then also \({{\mathrm{\mathcal{A}}\,}}\rightarrow {{\mathrm{\mathcal{A}}\,}}'\).

Proof

Suppose \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\). Let \(P\in {{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}}[t^{\pm 1}])\) and \(Q\in {{\mathrm{GL}\,}}_m({{{\mathbb {C}}\,}}[t^{\pm 1}])\) be matrices such that

$$\begin{aligned} (Y_1,\ldots ,Y_m):= (PX_1P^\top ,\ldots ,PX_mP^\top )Q \end{aligned}$$

is a tuple of matrices with coefficients in \({{{\mathbb {C}}\,}}[t]\) and \(X'_i=\lim _{t\rightarrow 0}Y_i\) for all \(i\in \{1,\ldots ,m\}\). Then it follows that \({{\mathrm{\mathcal{A}}\,}}'=\lim _{t\rightarrow 0}Q\cdot {{\mathrm{\mathcal{A}}\,}}\) and hence \({{\mathrm{\mathcal{A}}\,}}\rightarrow {{\mathrm{\mathcal{A}}\,}}'\). \(\square\)

6.2 The determinantal obstruction

Definition 6.2

The determinant of \({{{\mathcal{L}}\,}}\) is defined as

$$\begin{aligned} \det ({{{\mathcal{L}}\,}}):=\det (x_1X_1+\cdots +x_mX_m)\in {{{\mathbb {C}}\,}}[x_1,\ldots ,x_m]_n. \end{aligned}$$

When \((Y_1,\ldots ,Y_m)=(X_1,\ldots ,X_m)Q\) for some \(Q\in {{\mathrm{GL}\,}}_m({{{\mathbb {C}}\,}})\), then

$$\begin{aligned} y_1Y_1+\cdots +y_mY_m = x_1X_1+\cdots +x_mX_m \end{aligned}$$

for \((x_1,\ldots ,x_m)=(y_1,\ldots ,y_m)Q^{-\top }\). Hence \(\det ({{{\mathcal{L}}\,}})\) is well-defined up to coordinate change.‚ô¶

Since \({{{\mathcal{L}}\,}}\) is regular, its determinant is not the zero polynomial. Note that congruent subspaces have the same determinant up to scaling.

Definition 6.3

Let \(f,g\in {{{\mathbb {C}}\,}}[x_1,\ldots ,x_m]_n\) be forms. We say that f degenerates to g, denoted as \(f\rightarrow g\), when \(g\in \overline{{{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}})\cdot f}\).‚ô¶

Proposition 6.4

If \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\), then \(\det ({{{\mathcal{L}}\,}})\rightarrow \det ({{{\mathcal{L}}\,}}')\).

Proof

Suppose \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\). Let \(P\in {{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}}[t^{\pm 1}])\) and \(Q\in {{\mathrm{GL}\,}}_m({{{\mathbb {C}}\,}}[t^{\pm 1}])\) be matrices such that

$$\begin{aligned} (Y_1,\ldots ,Y_m):= (PX_1P^\top ,\ldots ,PX_mP^\top )Q \end{aligned}$$

is a tuple of matrices with coefficients in \({{{\mathbb {C}}\,}}[t]\) and \(X'_i=\lim _{t\rightarrow 0}Y_i\) for all \(i\in \{1,\ldots ,m\}\). Then

$$\begin{aligned} \det ({{{\mathcal{L}}\,}}')=\det (x_1X_1'+\cdots +x_mX_m')=\lim _{t\rightarrow 0}\det (x_1Y_1+\cdots +x_mY_m)=\lim _{t\rightarrow 0}\det (P{{{\mathcal{L}}\,}}P^\top ) \end{aligned}$$

and hence \(\det ({{{\mathcal{L}}\,}})\rightarrow \det ({{{\mathcal{L}}\,}}')\). \(\square\)

6.3 The rank-minimal subspace obstruction

Definition 6.5

We define the minimal matrix rank of \({{{\mathcal{L}}\,}}\) to be \(\tau _1({{{\mathcal{L}}\,}}):=\min \{{{\mathrm{rk}\,}}(X)\mid X\in {{{\mathcal{L}}\,}}\setminus \{\mathbf{0}_n\}\}\). More generally, for \(1\le k\le m\), we define

$$\begin{aligned} \tau _k({{{\mathcal{L}}\,}}):=\min \{\max \{{{\mathrm{rk}\,}}(X)\mid X\in {{\mathrm{\mathcal{P}}\,}}\setminus \{\mathbf{0}_n\}\}\mid {{\mathrm{\mathcal{P}}\,}}\in {{\mathrm{Gr}\,}}(k,{{{\mathcal{L}}\,}})\} \end{aligned}$$

to be the minimal upperbound on the rank of nonzero elements of a k-dimensional subspace of \({{{\mathcal{L}}\,}}\).‚ô¶

When \(k=2,3,4\), we call \(\tau _k({{{\mathcal{L}}\,}})\) the minimal pencil/net/web rank of \({{{\mathcal{L}}\,}}\).

Proposition 6.6

If \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\), then \(\tau _k({{{\mathcal{L}}\,}}')\le \tau _k({{{\mathcal{L}}\,}})\) for all \(1\le k\le m\).

Proof

This holds since the set

$$\begin{aligned} \{({{{\mathcal{L}}\,}},{{\mathrm{\mathcal{P}}\,}})\in {{\mathrm{Gr}\,}}(m,\mathbb{S}^n)\times {{\mathrm{Gr}\,}}(k,\mathbb{S}^n)\mid {{\mathrm{\mathcal{P}}\,}}\subseteq {{{\mathcal{L}}\,}}, \forall X\in {{\mathrm{\mathcal{P}}\,}}:{{\mathrm{rk}\,}}(X)\le \ell \} \end{aligned}$$

is closed for all \(\ell \ge 0\). \(\square\)

6.4 The Segre symbol obstruction

Let A be an \(n\times n\) matrix in Jordan normal form. The Segre symbol corresponding to A is a multiset of partitions

$$\begin{aligned} \sigma =\{(k^{(1)}_1,\ldots ,k^{(1)}_{n_1}),\ldots ,(k^{(\ell )}_1,\ldots ,k^{(\ell )}_{n_\ell }\} \end{aligned}$$

where A has \(\ell\) distinct eigenvalues \(\lambda _1,\ldots ,\lambda _\ell\) and has Jordan blocks of sizes \(k^{(i)}_1,\ldots ,k^{(i)}_{n_i}\) corresponding to \(\lambda _i\). In particular, we have \(\sum _{i,j}k^{(i)}_{j}=n\). Denote by \(\ge\) the partial order on the set of Segre symbols of \(n\times n\) matrices in Jordan normal form generated by

$$\begin{aligned} \{\tau _1,\ldots ,\tau _{\ell -1},\tau _\ell ,\tau _{\ell +1}\}\ge \{\tau _1,\ldots ,\tau _{\ell -1},\tau _\ell +\tau _{\ell +1}\} \end{aligned}$$

and

$$\begin{aligned} \{\tau _1,\ldots ,\tau _{\ell },\mu \}\ge \{\tau _1,\ldots ,\tau _\ell ,\mu '\} \hbox { when } \mu \triangleright \mu ' \end{aligned}$$

where \(\triangleright\) is the dominance order on partitions and the sum \((k_1,\ldots ,k_n)+(\ell _1,\ldots ,\ell _m)\) is defined to be

$$\begin{aligned} (k_1+\ell _1,\ldots ,k_{\max (n,m)}+\ell _{\max (n,m)}) \end{aligned}$$

with \(k_i=0\) for \(i>n\) and \(\ell _j=0\) for \(j>m\). See [6] for the history of the term Segre symbol.

Proposition 6.7

Let \({{{\mathcal{L}}\,}}\subseteq \mathbb{S}^n\) be a Jordan space and let \(U,V\in {{{\mathcal{L}}\,}}\) be invertible matrices. Then the following statements hold:

  1. (1)

    The sets of Segre symbols corresponding to the Jordan normal forms of matrices in \({{{\mathcal{L}}\,}}U^{-1}\) and in \({{{\mathcal{L}}\,}}V^{-1}\) are equal.

  2. (2)

    The set of Segre symbols corresponding to the Jordan normal form of a matrix in \({{{\mathcal{L}}\,}}U^{-1}\) has a unique maximal element.

Proof

  1. (1)

    Since the Jordan normal form of a matrix and its transpose are the same, it suffices to prove that \({{{\mathcal{L}}\,}}V^{-1}\) is similar to \(({{{\mathcal{L}}\,}}U^{-1})^\top = U^{-1}{{{\mathcal{L}}\,}}\). By [1, Lemma 2.3], we have \(WU^{-1}W=V\) for some matrix \(W\in {{{\mathcal{L}}\,}}\). And by Theorem 1.1(c) applied to \({{{\mathcal{L}}\,}}^{-1}\) and afterward to \({{{\mathcal{L}}\,}}\), we have

    $$\begin{aligned} {{{\mathcal{L}}\,}}=({{{\mathcal{L}}\,}}^{-1})^{-1}=V({{{\mathcal{L}}\,}}^{-1})V=VW^{-1}{{{\mathcal{L}}\,}}W^{-1} V. \end{aligned}$$

    Hence \({{{\mathcal{L}}\,}}V^{-1}=VW^{-1}{{{\mathcal{L}}\,}}W^{-1}\) is indeed similar to \(W^{-1}VW^{-1}{{{\mathcal{L}}\,}}=U^{-1}{{{\mathcal{L}}\,}}\).

  2. (2)

    Suppose that \(\sigma _1,\ldots ,\sigma _k\) are the maximal Segre symbols corresponding to the Jordan normal form of a matrix in \({{{\mathcal{L}}\,}}U^{-1}\). Then we see that

    $$\begin{aligned} {{{\mathcal{L}}\,}}U^{-1}=\bigcup _{i=1}^k\{ X\in {{{\mathcal{L}}\,}}U^{-1}\mid \hbox {the Segre symbol of} X \hbox {is at most} \sigma _i\} \end{aligned}$$

    Since \({{{\mathcal{L}}\,}}U^{-1}\) is subspace of \({{{\mathbb {C}}\,}}^{n\times n}\), it is irreducible. Since each of the sets on the right-hand side is closed, k must be equal to 1.

\(\square\)

Definition 6.8

We define the Segre symbol \(\sigma ({{{\mathcal{L}}\,}})\) of a Jordan space \({{{\mathcal{L}}\,}}\subseteq \mathbb{S}^n\) to be the maximal Segre symbol corresponding to the Jordan normal form of a matrix in \({{{\mathcal{L}}\,}}U^{-1}\) for any invertible matrix \(U\in {{{\mathcal{L}}\,}}\).‚ô¶

Remark 6.9

The set of Segre symbols corresponding to Jordan normal form of a matrix in \({{{\mathcal{L}}\,}}U^{-1}\) depends on U for general linear spaces of symmetric matrices. For example, let

$$\begin{aligned} {{{\mathcal{L}}\,}}=\left\{ \begin{pmatrix} x &{}z &{}y\\ z&{}x+y&{}0\\ y&{}0&{}x \end{pmatrix}\right\} . \end{aligned}$$

Then \({{{\mathcal{L}}\,}}\) does not contain nilpotent matrices. However, \(\mathbf{J}_3\in {{{\mathcal{L}}\,}}\) and

$$\begin{aligned} \left\{ \begin{pmatrix} 0&{}1&{}0\\ 0&{}0&{}1\\ 0&{}0&{}0 \end{pmatrix}\right\} \in {{{\mathcal{L}}\,}}\mathbf{J}_3 \end{aligned}$$

is nilpotent. Hence, the sets of Segre symbols are not the same.‚ô£

Question 6.10

Is the Segre symbol of a general linear space of symmetric matrices well-defined?

Proposition 6.11

Congruent Jordan spaces have the same Segre symbol.

Proof

Let \({{{\mathcal{L}}\,}}\subseteq \mathbb{S}^n\) be a Jordan space and \(U\in {{{\mathcal{L}}\,}},P\in {{\mathrm{GL}\,}}_n\) invertible matrices. Then \({{{\mathcal{L}}\,}}U^{-1}\) is similar to \(P{{{\mathcal{L}}\,}}U^{-1}P^{-1}=(P{{{\mathcal{L}}\,}}P^\top )(PUP^\top )^{-1}\) and hence \(\sigma ({{{\mathcal{L}}\,}})=\sigma (P{{{\mathcal{L}}\,}}P^\top )\). \(\square\)

Proposition 6.12

If \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\), then \(\sigma ({{{\mathcal{L}}\,}})\ge \sigma ({{{\mathcal{L}}\,}}')\).

Proof

Suppose that \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\). Let \(U'\in {{{\mathcal{L}}\,}}'\) an invertible matrix and \(X'\in {{{\mathcal{L}}\,}}'\) any other matrix. Then \((U',X')\) is the limit of a sequence of pairs \((U_n,X_n)\) of matrices contained in a Jordan space \({{{\mathcal{L}}\,}}_n\) congruent to \({{{\mathcal{L}}\,}}\). By replacing the sequence by a subsequence, we may assume that \(U_n\) is invertible. Since \(XU^{-1}=\lim _{n\rightarrow \infty }X_nU_n^{-1}\) and the Segre symbol corresponding to the Jordan normal form of \(X_nU_n^{-1}\) is at most \(\sigma ({{{\mathcal{L}}\,}})\), we see that the same holds for the Segre symbol corresponding to the Jordan normal form of \(XU^{-1}\). Hence \(\sigma ({{{\mathcal{L}}\,}}')\le \sigma ({{{\mathcal{L}}\,}})\). \(\square\)

6.5 The orbit dimension obstruction

Definition 6.13

We define the orbit dimension \(d({{{\mathcal{L}}\,}})\) of \({{{\mathcal{L}}\,}}\) to be the dimension of the irreducible variety \(\overline{\{(Y_1,\ldots ,Y_m)\mid {{\mathrm{span}\,}}(Y_1,\ldots ,Y_m)\in {{\mathrm{GL}\,}}_n\cdot {{{\mathcal{L}}\,}}\}}\).‚ô¶

Proposition 6.14

If \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\), then \(d({{{\mathcal{L}}\,}})\ge d({{{\mathcal{L}}\,}}')\) with equality if and only if \({{{\mathcal{L}}\,}},{{{\mathcal{L}}\,}}'\) are congruent.

Proof

Assume that \({{{\mathcal{L}}\,}}\rightarrow {{{\mathcal{L}}\,}}'\). Then

$$\begin{aligned} \overline{\{(Y_1,\ldots ,Y_m)\mid {{\mathrm{span}\,}}(Y_1,\ldots ,Y_m)\in {{\mathrm{GL}\,}}_n\cdot {{{\mathcal{L}}\,}}\}}\supseteq \overline{\{(Y_1,\ldots ,Y_m)\mid {{\mathrm{span}\,}}(Y_1,\ldots ,Y_m)\in {{\mathrm{GL}\,}}_n\cdot {{{\mathcal{L}}\,}}'\}} \end{aligned}$$

and hence \(d({{{\mathcal{L}}\,}})\ge d({{{\mathcal{L}}\,}}')\). Since both varieties are irreducible, equality of the dimensions implies that \({{{\mathcal{L}}\,}},{{{\mathcal{L}}\,}}'\) are congruent. \(\square\)

7 Families of degenerations of Jordan nets

Fix an integer \(n\ge 2\). Theorem 3.3 classifies all embeddings of the Jordan algebras \({{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\times {{{\mathbb {C}}\,}}\), \({{{\mathbb {C}}\,}}\times {{\mathrm{\mathcal{J}}\,}}^1_0\), \({{{\mathbb {C}}\,}}[x]/(x^3)\), \({{\mathrm{\mathcal{J}}\,}}^2_2\) and \({{\mathrm{\mathcal{J}}\,}}^2_1\) into \(\mathbb{S}^n\). The goal of this section is to determine all degenerations between these embeddings. The abstract obstruction shows that there are no degenerations between the A’s and the B’s. So we can handle them separately. For the construction of the degenerations, see Appendix 1. We now prove that these degeneration generate everything.

Proposition 7.1

The degenerations

\(\mathrm {(a)}\):

\(A^{(1)}_{k_1,k_2,k_3}\rightarrow A^{(2)}_{k_3,k_1,k_2+k_3}\)

\(\mathrm {(b)}\):

\(A^{(1)}_{k_1,k_2,k_3}\rightarrow A^{(2)}_{k_2+k_3,k_1+k_2,k_3}\)

\(\mathrm {(c)}\):

\(A^{(1)}_{k_1,k_2,k_3}\rightarrow A^{(2)}_{k_1+k_2+k_3,k_2,k_3}\)

\(\mathrm {(d)}\):

\(A^{(2)}_{r,k_1,k_2}\rightarrow A^{(2)}_{r,k_1+2,k_2-1}\) for \(k_2>1\)

\(\mathrm {(e)}\):

\(A^{(2)}_{r,k_1,k_2}\rightarrow A^{(3)}_{k_1,k_2-r,r}\) for \(r\le k_2\)

\(\mathrm {(f)}\):

\(A^{(2)}_{r,k_1,k_2}\rightarrow A^{(3)}_{k_1+k_2-r,r-k_2,k_2}\) for \(k_2\le r\le k_1+k_2\)

\(\mathrm {(g)}\):

\(A^{(2)}_{r,k_1,k_2}\rightarrow A^{(3)}_{r-(k_1+k_2),k_1,k_2}\) for \(r\ge k_1+k_2\)

\(\mathrm {(h)}\):

\(A^{(3)}_{k_1,k_2,k_3}\rightarrow A^{(3)}_{k_1+1,k_2+1,k_3-1}\) for \(k_3>1\)

\(\mathrm {(i)}\):

\(A^{(3)}_{k_1,k_2,k_3}\rightarrow A^{(3)}_{k_1+2,k_2-1,k_3}\) for \(k_2>0\)

\(\mathrm {(j)}\):

\(A^{(3)}_{k_1,k_2,k_3}\rightarrow A^{(3)}_{k_1-1,k_2+2,k_3-1}\) for \(k_1>0,k_3>1\)

generate all degenerations between Jordan nets labeled with an A.

Proof

The following table shows for each orbit its determinant, minimal rank, minimal pencil rank and Segre symbol.

$$\begin{aligned} \begin{array}{l|c|c|c|c|} &{}\det &{}\tau _1&{}\tau _2&{}\sigma \\ \hline A^{(1)}_{k_1,k_2,k_3}&{}x^{k_1+k_2+k_3}y^{k_2+k_3}z^{k_3}&{}k_3&{}k_2+2k_3&{}(\overbrace{1\cdots \cdot \cdot 1}^{k_1+k_2+k_3})(\overbrace{1\cdots \cdot 1}^{k_2+k_3})(\overbrace{1\cdots 1}^{k_3})\\ \hline A^{(2)}_{r,k_1,k_2}&{}x^ry^{k_1+2k_2}&{}\min (r,k_2)&{}\min (r+k_2,k_1+2k_2)&{}(\overbrace{1\cdots 1}^{r})(\overbrace{2\cdots 2}^{k_2}\overbrace{1\cdots 1}^{k_1})\\ \hline A^{(3)}_{k_1,k_2,k_3}&{}x^n&{}k_3&{}k_2+2k_3&{}(\overbrace{3\cdots 3}^{k_3}\overbrace{2\cdots 2}^{k_2}\overbrace{1\cdots 1}^{k_1})\\ \hline \end{array} \end{aligned}$$

The abstract obstruction shows that we have the following six possible cases:

(1) Suppose that \(A^{(3)}_{k_1,k_2,k_3}\rightarrow A^{(3)}_{\ell _1,\ell _2,\ell _3}\). Then the Segre symbol obstruction shows that

$$\begin{aligned} (\overbrace{3\cdots 3}^{\ell _3}\overbrace{2\cdots 2}^{\ell _2}\overbrace{1\cdots 1}^{\ell _1})\le (\overbrace{3\cdots 3}^{k_3}\overbrace{2\cdots 2}^{k_2}\overbrace{1\cdots 1}^{k_1}) \end{aligned}$$

This means that we can reach \((\ell _1,\ell _2,\ell _3)\) from \((k_1,k_2,k_3)\) by a series of moves where in each step we replace \((k_1,k_2,k_3)\) by either \((k_1+1,k_2+1,k_3-1)\), \((k_1+2,k_2-1,k_3)\) or \((k_1-1,k_2+2,k_3-1)\). Hence \(A^{(3)}_{k_1,k_2,k_3}\rightarrow A^{(3)}_{\ell _1,\ell _2,\ell _3}\) is obtained as a composition of degenerations from (h),(i),(j).

(2) Suppose that \(A^{(2)}_{r,k_1,k_2}\rightarrow A^{(3)}_{\ell _1,\ell _2,\ell _3}\). The Segre symbol obstruction shows that

$$\begin{aligned} (\overbrace{3\cdots 3}^{\ell _3}\overbrace{2\cdots 2}^{\ell _2}\overbrace{1\cdots 1}^{\ell _1})\le \left\{ \begin{array}{ll} (\overbrace{3\cdots 3}^{r}\overbrace{2\cdots 2}^{k_2-r}\overbrace{1\cdots 1}^{k_1})&{}\mathrm{if} \quad r\le k_2\\ (\overbrace{3\cdots 3}^{k_2}\overbrace{2\cdots 2}^{r-k_2}\overbrace{1\cdots \cdot 1}^{k_1+k_2-r})&{}\mathrm{if} \quad k_2\le r\le k_1+k_2\\ (\overbrace{3\cdots 3}^{k_2}\overbrace{2\cdots 2}^{k_1}\overbrace{1\cdots \cdots 1}^{r-(k_1+k_2)})&{}\mathrm{if} \quad r\ge k_1+k_2 \end{array}\right. \end{aligned}$$

and hence we have:

  • \(A^{(3)}_{k_1,k_2-r,r}\rightarrow A^{(3)}_{\ell _1,\ell _2,\ell _3}\) when \(r\le k_2\);

  • \(A^{(3)}_{k_1+k_2-r,r-k_2,k_2}\rightarrow A^{(3)}_{\ell _1,\ell _2,\ell _3}\) when \(k_2\le r\le k_1+k_2\); and

  • \(A^{(3)}_{r-(k_1+k_2),k_1,k_2}\rightarrow A^{(3)}_{\ell _1,\ell _2,\ell _3}\) when \(r\ge k_1+k_2\)

by (1). So \(A^{(2)}_{r,k_1,k_2}\rightarrow A^{(3)}_{\ell _1,\ell _2,\ell _3}\) is obtained as a composition of degenerations from (1),(e),(f),(g).

(3) Suppose that \(A^{(2)}_{r,k_1,k_2}\rightarrow A^{(2)}_{r',\ell _1,\ell _2}\). Then the determinantal obstruction shows that \(\{r,k_1+2k_2\}=\{r',\ell _1+2\ell _2\}\). Now, the Segre symbol obstruction shows that \(r=r'\) and \(\ell _2\le k_2\). Hence the degeneration is from (d).

(4) Suppose that \(A^{(1)}_{k_1,k_2,k_3}\rightarrow A^{(3)}_{\ell _1,\ell _2,\ell _3}\). Then the Segre symbol obstruction shows that

$$\begin{aligned} (\overbrace{3\cdots 3}^{\ell _3}\overbrace{2\cdots 2}^{\ell _2}\overbrace{1\cdots 1}^{\ell _1})\le (\overbrace{3\cdots 3}^{k_3}\overbrace{2\cdots 2}^{k_2}\overbrace{1\cdots 1}^{k_1}) \end{aligned}$$

and hence \(A^{(3)}_{k_1,k_2,k_3}\rightarrow A^{(3)}_{\ell _1,\ell _2,\ell _3}\). We have \(A^{(1)}_{k_1,k_2,k_3}\rightarrow A^{(3)}_{k_1,k_2,k_3}\) using for example (a),(e). So using (1), we are done.

(5) Suppose that \(A^{(1)}_{k_1,k_2,k_3}\rightarrow A^{(2)}_{r,\ell _1,\ell _2}\). Then the Segre symbol obstruction shows that

$$\begin{aligned} (\overbrace{1\cdots 1}^{r})(\overbrace{2\cdots 2}^{\ell _2}\overbrace{1\cdots 1}^{\ell _1})\le (\overbrace{1\cdots \cdot \cdot 1}^{k_1+k_2+k_3})(\overbrace{1\cdots \cdot 1}^{k_2+k_3})(\overbrace{1\cdots 1}^{k_3}) \end{aligned}$$

and hence \(r\in \{k_1+k_2+k_3,k_2+k_3,k_3\}\). When \(r=k_3\), we see that

$$\begin{aligned} (\overbrace{2\cdots 2}^{\ell _2}\overbrace{1\cdots 1}^{\ell _1})\le (\overbrace{1\cdots \cdot \cdot 1}^{k_1+k_2+k_3})(\overbrace{1\cdots \cdot 1}^{k_2+k_3}) \end{aligned}$$

which implies that \(\ell _2\le k_2+k_3\) and so \(A^{(2)}_{k_3,k_1,k_2+k_3}\rightarrow A^{(2)}_{r,\ell _1,\ell _2}\) using (a),(d). When \(r>k_3\), the minimal rank obstruction shows that \(\ell _2\le k_3\) and so \(A^{(2)}_{r,n-r-2k_3,k_3}\rightarrow A^{(2)}_{r,\ell _1,\ell _2}\) using (b),(d) or (c),(d).

(6) Suppose that \(A^{(1)}_{k_1,k_2,k_3}\rightarrow A^{(1)}_{\ell _1,\ell _2,\ell _3}\). Then the determinant obstruction shows that

$$\begin{aligned} x^{k_1+k_2+k_3}y^{k_2+k_3}z^{k_3}\rightarrow x^{\ell _1+\ell _2+\ell _3}y^{\ell _2+\ell _3}z^{\ell _3} \end{aligned}$$

which implies that \((k_1,k_2,k_3)=(\ell _1,\ell _2,\ell _3)\). \(\square\)

Proposition 7.2

The degenerations

\(\mathrm {(a)}\):

\(B^{(1)}_{n/2}\rightarrow B^{(2)}_{k,0,n/2}\) for \(1\le k\le n/4\) if \(2\mid n\)

\(\mathrm {(b)}\):

\(B^{(2)}_{k,\ell _1,\ell _2}\rightarrow B^{(2)}_{k-1,\ell _1,\ell _2}\) for \(k>1\)

generate all degenerations between Jordan nets labeled with an B.

Proof

The following table shows for each orbit its determinant, minimal rank and minimal pencil rank.

$$\begin{aligned} \begin{array}{l|c|c|c|} &{}\det &{}\tau _1&{}\tau _2\\ \hline B^{(1)}_{n/2}&{}(xy-z^2)^{n/2}&{}n/2&{}n\\ \hline B^{(2)}_{k,\ell _1,\ell _2}&{}x^{\ell _2}y^{\ell _1+\ell _2}&{}2k&{}\ell _2\\ \hline \end{array} \end{aligned}$$

The abstract obstruction shows that we have the following two possible cases:

(1) Suppose that \(B^{(2)}_{k,\ell _1,\ell _2}\rightarrow B^{(2)}_{k',\ell _1',\ell _2'}\). Then the determinantal obstruction shows that \((\ell _1,\ell _2)=(\ell _1',\ell _2')\) and the minimum rank condition shows that \(k'\le k\). So the degenerations is from (b).

(2) Suppose that \(B^{(1)}_{n/2}\rightarrow B^{(2)}_{k,\ell _1,\ell _2}\). Then the determinantal obstruction shows that \((\ell _1,\ell _2)=(0,n/2)\) and the minimum rank obstruction shows that \(k\le n/4\). So the degenerations is from (a). \(\square\)

We see that the closures of the orbits \(A^{(1)}_{k_1,k_2,k_3}\) and \(B^{(1)}_{n/2}\) are always components of the Jordan locus. We compute their codimensions in \({{\mathrm{Gr}\,}}(3,\mathbb{S}^n)\).

Proposition 7.3

Write \((n_1,n_2,n_3)=(k_1+k_2+k_3,k_2+k_3,k_3)\). Then \(A^{(1)}_{k_1,k_2,k_3}\) is invariant under

$$\begin{aligned} \left\{ \begin{pmatrix}\lambda _1Q_1\\ {} &{}\lambda _2Q_2\\ &{}&{}\lambda _3Q_3\end{pmatrix}\,\bigg {|}\,\begin{array}{l}Q_1\in O(n_1),\lambda _1\in {{{\mathbb {C}}\,}}^*,\\ Q_2\in O(n_2),\lambda _2\in {{{\mathbb {C}}\,}}^*,\\ Q_3\in O(n_3),\lambda _3\in {{{\mathbb {C}}\,}}^*\end{array} \right\} \end{aligned}$$

together with

$$\begin{aligned} P_1:=\begin{pmatrix}&{}\mathbf{1}_{n_1}\\ \mathbf{1}_{n_1}\\ &{}&{}\mathbf{1}_{n_3}\end{pmatrix}\hbox { and }P_2:=\begin{pmatrix}\mathbf{1}_{n_1}\\ &{}&{}\mathbf{1}_{n_2}\\ {} &{}\mathbf{1}_{n_2}\end{pmatrix} \end{aligned}$$

when \(n_1=n_2\) and when \(n_2=n_3\), respectively. These matrices generate the stabilizer of \(A^{(1)}_{k_1,k_2,k_3}\). The orbit of \(A^{(1)}_{k_1,k_2,k_3}\) has codimension \(n_1^2 + n_1n_2 + n_2^2 + n_1n_3 + n_2n_3 + n_3^2 + n_1 + n_2 + n_3 - 6\).

Proof

Let

$$\begin{aligned} \begin{array}{rrr} A\in {{{\mathbb {C}}\,}}^{n_1\times n_1},&{}B\in {{{\mathbb {C}}\,}}^{n_1\times n_2},&{}C\in {{{\mathbb {C}}\,}}^{n_1\times n_3},\\ D\in {{{\mathbb {C}}\,}}^{n_2\times n_1},&{}E\in {{{\mathbb {C}}\,}}^{n_2\times n_2},&{}F\in {{{\mathbb {C}}\,}}^{n_2\times n_3},\\ G\in {{{\mathbb {C}}\,}}^{n_3\times n_1},&{}H\in {{{\mathbb {C}}\,}}^{n_3\times n_2},&{}I\in {{{\mathbb {C}}\,}}^{n_3\times n_3} \end{array} \end{aligned}$$

be matrices such that

$$\begin{aligned} \begin{pmatrix} A&{}B&{}C\\ D&{}E&{}F\\ G&{}H&{}I \end{pmatrix}A^{(1)}_{k_1,k_2,k_3}\begin{pmatrix} A&{}B&{}C\\ D&{}E&{}F\\ G&{}H&{}I \end{pmatrix}^\top =A^{(1)}_{k_1,k_2,k_3}. \end{aligned}$$

Then

$$\begin{aligned} \begin{pmatrix} AA^\top &{}AD^\top &{}AG^\top \\ DA^\top &{}DD^\top &{}DG^\top \\ GA^\top &{}GD^\top &{}GG^\top \end{pmatrix}=\begin{pmatrix} A&{}B&{}C\\ D&{}E&{}F\\ G&{}H&{}I \end{pmatrix}\begin{pmatrix}{} \mathbf{1}_{n_1}\\ {} &{}\mathbf{0}_{n_2}\\ &{}&{}\mathbf{0}_{n_3}\end{pmatrix} \begin{pmatrix} A&{}B&{}C\\ D&{}E&{}F\\ G&{}H&{}I \end{pmatrix}^\top \in A^{(1)}_{k_1,k_2,k_3}. \end{aligned}$$

So of A, D, G, we see that one is of the form \(\lambda _1 Q_1\) with \(Q_1\in O(n_1)\) and \(\lambda _1\in {{{\mathbb {C}}\,}}^*\). It then easily follows that the other two matrices are zero. Note that \(D=\lambda _1 Q_1\) is only possible when \(n_1=n_2\) and \(G=\lambda Q_1\) is only possible when \(n_1=n_2=n_3\). Similarly, we see that one of B, E, H is of the form \(\lambda _2 Q_2\) with \(Q_2\in O(n_2)\) and \(\lambda _2\in {{{\mathbb {C}}\,}}^*\) and the other two matrices are zero. And, we see that one of C, F, I is of the form \(\lambda _2 Q_2\) with \(Q_3\in O(n_3)\) and \(\lambda _3\in {{{\mathbb {C}}\,}}^*\) and the other two matrices are zero. It is straightforward to check that this matrix is in the group generated by the given matrices. \(\square\)

Proposition 7.4

The stabilizer of \(B^{(1)}_{n/2}\) is

$$\begin{aligned} \left\{ \begin{pmatrix}aQ&{}bQ\\ cQ&{}dQ\end{pmatrix}\,\bigg {|}\,Q\in O(n/2),\begin{pmatrix}a&{}b\\ c&{}d\end{pmatrix}\in {{\mathrm{GL}\,}}_2({{{\mathbb {C}}\,}})\right\} . \end{aligned}$$

The orbit of \(B^{(1)}_{n/2}\) has codimension \(5(n^2/8 + n/4 - 1)\).

Proof

Let \(A,B,C,D\in {{{\mathbb {C}}\,}}^{n/2\times n/2}\) be such that

$$\begin{aligned} \begin{pmatrix}A&{}B\\ C&{}D\end{pmatrix}B^{(1)}\begin{pmatrix}A&{}B\\ C&{}D\end{pmatrix}^\top =B^{(1)}. \end{aligned}$$

Then

$$\begin{aligned} \begin{pmatrix}AA^\top &{}AC^\top \\ CA^\top &{}CC^\top \end{pmatrix}=\begin{pmatrix}A&{}B\\ C&{}D\end{pmatrix}\begin{pmatrix}\mathbf{1}_{n/2}\\ {} &{}\mathbf{0}_{n/2}\end{pmatrix}\begin{pmatrix}A&{}B\\ C&{}D\end{pmatrix}^\top \in B^{(1)} \end{aligned}$$

and hence \((A,C)=(a Q,c Q)\) for some \(Q\in O(n/2)\) and \(a,c\in {{{\mathbb {C}}\,}}\). Similarly, we find that \((B,D)=(b P,d P)\) for some \(P\in O(n/2)\) and \(b,d\in {{{\mathbb {C}}\,}}\). Now

$$\begin{aligned} g:=\begin{pmatrix}A&{}B\\ C&{}D\end{pmatrix}=\begin{pmatrix}a\mathbf{1}_{n/2}&{}b\mathbf{1}_{n/2}\\ c\mathbf{1}_{n/2}&{}d\mathbf{1}_{n/2}\end{pmatrix}\begin{pmatrix}Q\\ {} &{}P\end{pmatrix} \end{aligned}$$

and so \({{\mathrm{Diag}\,}}(Q,P)\) also lies in the stabilizer of \(B^{(1)}_{n/2}\). It is straightforward to check that this is only possible when P, Q are linearly dependent. So g must be of the required form. \(\square\)

We believe that the codimensions of the other orbits are also polynomials in their parameters.

Conjecture 7.5

The codimensions of \(A^{(2)}_{r,k_1,k_2},A^{(3)}_{k_1,k_2,k_3},B^{(2)}_{k,\ell _1,\ell _2}\) in \({{\mathrm{Gr}\,}}(3,\mathbb{S}^n)\) are polynomials functions \(f_2(r,k_1,k_2),f_3(k_1,k_2,k_3),g(k,\ell _1,\ell _2)\) of degree \(\le 2\), respectively.

Assuming the conjecture, we find

$$\begin{aligned} f_2(r,k_1,k_2)&= {} r^2 + rk_1 + k_1^2 + 2rk_2 + 3k_1k_2 + 3k_2^2 + r + k_1 + 2k_2 - 5,\\ f_3(k_1,k_2,k_3)&= {} k_1^2 + 3k_1k_2 + 3k_2^2 + 4k_1k_3 + 8k_2k_3 + 6k_3^2 + k_1 + 2k_2 + 3k_3 - 4,\\ g(k,\ell _1,\ell _2)&= {} 3\ell _1^2/2 + 2\ell _1\ell _2 + 5\ell _2^2/2 - \ell _1/2 + 5\ell _2/2 - 5. \end{aligned}$$

8 Proofs of the main results

In this section, we prove the results from Sect. 4.

Remark 8.1

To prove that Fig. 3 contains all degenerations, one needs to prove in particular that \(A^{(1)}_{1,0,1}\not \rightarrow B^{(2)}_{1,0,2}\) and \(A^{(2)}_{2,0,1}\not \rightarrow C_{4,1}\). In [1], this was proven using equations on the orbits of \(A^{(1)}_{1,0,1}\) and \(A^{(2)}_{2,0,1}\) that do not hold for \(B^{(2)}_{1,0,2}\) and \(C_{4,1}\), respectively, found by a computer search. Now, we also see that \(A^{(1)}_{1,0,1}\not \rightarrow B^{(2)}_{1,0,2}\) follows from the minimal rank obstruction and \(A^{(2)}_{2,0,1}\not \rightarrow C_{4,1}\) follows from the Segre symbol obstruction.♣

Proof of Theorem 4.3

To show that the diagram describes all degenerations, we need to show that \(A^{(2)}_{3,0,1},B^{(2)}_{1,1,2}\not \rightarrow C_{5,1}\). Note that the set

$$\begin{aligned} \left\{ ({{{\mathcal{L}}\,}},{{\mathrm{\mathcal{P}}\,}})\in {{\mathrm{Gr}\,}}(3,\mathbb{S}^n)\times {{\mathrm{Gr}\,}}(2,\mathbb{S}^n)\,\bigg {|}\, \begin{array}{c}{{\mathrm{\mathcal{P}}\,}}\subseteq {{{\mathcal{L}}\,}}, {{\mathrm{rk}\,}}(X)\le 2\hbox { for all }X\in {{\mathrm{\mathcal{P}}\,}},\\ \det (Q{{\mathrm{\mathcal{P}}\,}}Q^\top )\in \{f^2\}\hbox { for all }Q\in {{{\mathbb {C}}\,}}^{2\times n}\end{array}\right\} \end{aligned}$$

is closed and \({{\mathrm{GL}\,}}_n({{{\mathbb {C}}\,}})\)-stable. Hence so is its projection on \({{\mathrm{Gr}\,}}(3,\mathbb{S}^n)\). The orbits \(A^{(2)}_{3,0,1},B^{(2)}_{1,1,2}\) are contained in this projection while the orbit \(C_{5,1}\) is not. So indeed \(A^{(2)}_{3,0,1},B^{(2)}_{1,1,2}\not \rightarrow C_{5,1}\). \(\square\)

Proof of Proposition 4.5

Suppose that \(B^{(2)}_{1,0,3}\rightarrow C_{6,6}\) and identify \({{{\mathcal{L}}\,}}\subseteq \mathbb{S}^6\) with

$$\begin{aligned} (a,b,c,d,e,f){{{\mathcal{L}}\,}}(a,b,c,d,e,f)^\top \end{aligned}$$

so that \(B^{(2)}_{1,0,3}={{\mathrm{span}\,}}(2ac+b^2,ad,2df+e^2)\) and \(C_{6,6}={{\mathrm{span}\,}}(ab,ac,af+be+cd)\). Using Remark 4.2, there exist linear forms A, B, C, D, E, F in a, b, c, d, e, f over \({{{\mathbb {C}}\,}}((t))\) and a matrix \(Q\in {{\mathrm{GL}\,}}_3({{{\mathbb {C}}\,}}((t)))\) such that every entry of

$$\begin{aligned} (G_1,G_2,G_3):= (2AC+B^2,AD,2DF+E^2)Q \end{aligned}$$

is a form with coefficients in \({{{\mathbb {C}}\,}}[[t]]\) and \(\lim _{t\rightarrow 0}(G_1,G_2,G_2)=(ab,ac,af+be+cd)\). Note that

$$\begin{aligned} 2AC+B^2,AD,2DF+E^2 \end{aligned}$$

are linearly independent over \({{{\mathbb {C}}\,}}((t))\) and

$$\begin{aligned} \lambda _1(2AC+B^2)+\lambda _2AD+\lambda _3(DF+E^2), \quad \lambda _1,\lambda _2,\lambda _3\in {{{\mathbb {C}}\,}}((t)) \end{aligned}$$

converges to an element of \(C_6\) as \(t\rightarrow 0\) whenever its coefficients lie in \({{{\mathbb {C}}\,}}[[t]]\). Note that we are allowed to replace (A, B, C, D, E, F) by \((\mu _1A,\mu _2B,\mu _3C,\mu _4D,\mu _5E,\mu _6F)\) as long as \(\mu _1\mu _3=\mu _2^2\) and \(\mu _4\mu _6=\mu _5^2\). So we may assume that A, D converge to nonzero forms in a, b, c, d, e, f. Now AD converges to a nonzero element of \(C_6\). Since AD has rank 2, this element must be \(a(\lambda b+\mu c)\) for some \((\lambda :\mu )\in {{\mathrm{\mathbb {P}}\,}}^1\). Using a base change in b, c, we may assume that \((\lambda ,\mu )=(1,0)\). Using symmetry and by scaling A, D, we may assume that \(A\rightarrow a\) and \(D\rightarrow b\) as \(t\rightarrow 0\). Next, consider the form \(2DF+E^2\). Since \(AD\rightarrow ab\) as \(t\rightarrow 0\), there exists an \(\lambda \in {{{\mathbb {C}}\,}}((t))\) such that

$$\begin{aligned} \lambda AD+2DF+E^2\in {{\mathrm{span}\,}}_{{{{\mathbb {C}}\,}}((t))}(a^2,b^2,ac,bc,c^2) \end{aligned}$$

and we replace F by \(F+\lambda D/2\). We scale E, F such that \(2DF+E^2\) converges to a nonzero element of \(C_6\). Since \(2DF+E^2\) has rank 3 and its limit lies in \({{\mathrm{span}\,}}(a^2,b^2,ac,bc,c^2)\), we can scale so that \(2DF+E^2\rightarrow ac\) as \(t\rightarrow 0\).

We now see that (b, ac) is a limit of pairs of the form \((D,2DF+E^2)\) where D, E, F are forms in a, b, c, d, e, f. By setting d, e, f to zero, we see that (b, ac) is also a limit of pairs of the form \((D,2DF+E^2)\) where D, E, F are forms in a, b, c. The closure of such pairs forms a hyperplane in \({{{\mathbb {C}}\,}}\{a,b,c\}\times {{{\mathbb {C}}\,}}\{a^2,ab,b^2,ac,bc,c^2\}\) that does not contain (b, ac). This is a contradiction. \(\square\)

Proof of Theorem 4.6

We have the following obstructions:

  • We have \(B^{(2)}_{1,0,3}\not \rightarrow C_{6,6}\) by Proposition¬†4.5.

  • We have \(A^{(1)}_{3,0,1}\not \rightarrow C_{6,5}\) since \(\tau _2(A^{(1)}_{3,0,1})=2<3=\tau _2(C_{6,5})\).

  • We have \(A^{(1)}_{3,0,1}\not \rightarrow C_{6,6}\) since \(\tau _1(A^{(1)}_{3,0,1})=1<2=\tau _1(C_{6,6})\).

  • We have \(A^{(2)}_{3,1,1}\not \rightarrow C_{6,4}\) since \(\sigma (A^{(2)}_{3,1,1})=(111)(21)\not \ge (222)=\sigma (C_{6,4})\).

  • We have \(A^{(2)}_{4,0,1}\not \rightarrow C_{6,7}\) since \(\sigma (A^{(2)}_{4,0,1})=(1111)2\not \ge (2211)=\sigma (C_{6,7})\).

  • We have \(B^{(2)}_{1,2,2}\not \rightarrow C_{6,5}\) since \(\tau _2(B^{(2)}_{1,2,2})=2<3=\tau _2(C_{6,5})\).

  • We have \(A^{(3)}_{0,0,2}\not \rightarrow C_{6,4}\) since \(\sigma (A^{(3)}_{0,0,2})=(33)\not \ge (222)=\sigma (C_{6,4})\).

Using these obstructions, we see that the only possibly missing degenerations are the dotted lines. \(\square\)