# Formats of $6\times 6$ skew matrices of linear forms with vanishing Pfaffian

Christian Böhning  Christian Böhning, Mathematics Institute, University of
Warwick
Coventry CV4 7AL, England<EMAIL_ADDRESS>and  Hans-Christian Graf
von Bothmer  Hans-Christian Graf von Bothmer, Fachbereich Mathematik der
Universität Hamburg
Bundesstraße 55
20146 Hamburg, Germany<EMAIL_ADDRESS>
###### Abstract.

We show that every skew-symmetric $6\times 6$ matrix of linear forms with
vanishing Pfaffian is congruent to one of finitely many types of matrices,
each of which is characterised by a specific pattern of zeroes (and some other
linear relations) among its entries. Such matrices are for example important
for compactifying moduli spaces of stable rank $2$ vector bundles with Chern
classes $c_{1}=0,c_{2}=2$ on cubic threefolds.

The first author was supported by the EPSRC New Horizons Grant EP/V047299/1.

In this paper we prove that a skew-symmetric $6\times 6$ matrix of linear
forms with vanishing Pfaffian has at least one of the skew formats in Table 1.
This result is known by work of Manivel and Mezzetti [MaMe05] for matrices of
linear forms whose rank at most $2$ locus is empty. This condition implies, in
particular, that the entries of such a matrix span at most a $\mathbb{P}^{2}$.
Wo do not make any additional assumptions in this paper.

Our result is also known by work of Comaschi [Co21] if one assumes that the
matrix is stable. This condition implies that the span of the entries of such
matrices is at least a $\mathbb{P}^{3}$ and that formats $(a),(b)$ and $(c)$
do not occur. We extend Comaschi’s classification to the strictly semistable
case in which only format $(e)$ is possible.

In fact Manivel, Mezzetti and Comaschi prove more: they show that the matrices
of each format form an irreducible orbit under an appropriate group and even
give a normal form in each case. While it is clear that the matrices of each
format form an orbit of a projective space under the
$\mathrm{GL}_{6}(\mathbb{C})$ operation $A\mapsto SAS^{t}$ we do not attempt
to give normal forms in the general case. We do find normal forms in the
strictly semistable case, and also obtain a slight variation of Comaschi’s
normal forms for stable $M$ from our methods.

Notice that it is obvious that matrices of the formats given in Table 1 have
vanishing Pfaffians. The difficulty lies in proving that there are no others.

(a) | $\left(\begin{smallmatrix}0&*&*&*&*&\\\ *&0&*&*&*&\\\ *&*&0&*&*&\\\ *&*&*&0&*&\\\ *&*&*&*&0&\\\ \\\ \end{smallmatrix}\right)$
---|---
(b) | $\left(\begin{smallmatrix}&*&*&*&*&*\\\ *&&*&*&*&*\\\ *&*&\\\ *&*&\\\ *&*&\\\ *&*&\end{smallmatrix}\right)$
(c) | $\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&\\\ *&*&0&*&\\\ *&*&*&0&\\\ *\\\ *\end{smallmatrix}\right)$

(d) | $\left(\begin{smallmatrix}0&*&*&\\\ *&0&*&\\\ *&*&0&\\\ &&&0&*&*\\\ &&&*&0&*\\\ &&&*&*&0\end{smallmatrix}\right)$
---|---
(e) | $\left(\begin{smallmatrix}0&*&*&0&*&*\\\ *&0&*&*&0&*\\\ *&*&0&*&*&0\\\ 0&*&*\\\ *&0&*\\\ *&*&0\end{smallmatrix}\right)$
(f) | $\left(\begin{smallmatrix}&*&&0&*&*\\\ *&&&*&0&*\\\ &&&*&*&0\\\ 0&*&*&&*&\\\ *&0&*&*&&\\\ *&*&0&&&\end{smallmatrix}\right)$

Table 1. Formats of skew-symmetric $6\times 6$ matrices of linear forms with
vanishing Pfaffian. The last three are to be understood as double skew
formats.

##  1\. Formats

Let $V$ be a finite-dimensional $\mathbb{C}$-vector space, $\mathbb{P}(V)$ the
associated projective space. We say that $M$ is a matrix of linear forms over
$\mathbb{P}(V)$ if it has entries in $V^{*}$. For any matrix $M$ we denote the
entry in row $i$ and column $j$ by $m_{ij}$. We think of the spaces of rows
and columns as projective spaces and therefore let row and column indices
start with zero. Our notation for the span of these entries is $\langle
M\rangle:=\langle m_{ij}\rangle\subset V^{*}$. Notice that this span could be
a proper subspace of $V^{*}$.

For a skew-symmetric matrix $M$ we denote by $M_{ij}$ the matrix obtained by
deleting the $i$th and $j$th row and column of $M$. We also denote by
$\mathrm{Pf}(M)$ the Pfaffian of $M$ if $M$ is skew.

We now introduce the language of formats of matrices.

######  Definition 1.1 (Formats of matrices).

Let $M$ be an $r\times s$ matrix of linear forms over $\mathbb{P}(V)$ and $F$
an $r\times s$ matrix with entries $0$ or $*$.

  1. a)

We say that $M$ is of the form, or has form, $F$ if $m_{ij}$ is $0$ whenever
$f_{ij}$ is $0$. In particular, $M$ may also have $0$ entries where $F$ has
$*$ entries.

  2. b)

We say that $M$ has format $F$ if there exist matrices
$S\in\mathrm{GL}_{r}(\mathbb{C})$ and $T\in\mathrm{GL}_{s}(\mathbb{C})$ such
that $SMT$ hasr form $F$.

  3. c)

If $r=s$ and $M$ is skew, we say that $M$ has skew format $F$ if there exists
$S\in\mathrm{GL}_{r}(\mathbb{C})$ such that $SMS^{t}$ has form $F$.

Sometimes we replace a $0$ by a blank space if we feel that this improves the
readability.

######  Example 1.2.

$M=\left(\begin{smallmatrix}x&0\\\ y&0\end{smallmatrix}\right)$ has form
$\left(\begin{smallmatrix}*&0\\\ *&0\end{smallmatrix}\right)$.
$M^{\prime}=\left(\begin{smallmatrix}x&x\\\ y&y\end{smallmatrix}\right)$ has
format $\left(\begin{smallmatrix}*&0\\\ *&0\end{smallmatrix}\right)$, since it
can be transformed into $M$ by row and column operations.

######  Definition 1.3.

In this paper we consider $6\times 6$ skew matrices $M$ of linear forms over a
projective space $\mathbb{P}(V)$. In this special situation we say that $M$ is
double skew if there are skew symmetric $3\times 3$ matrices of linear forms
$N_{0},N_{1},N_{2}$ over $\mathbb{P}(V)$ such that

$M=\begin{pmatrix}N_{0}&N_{1}\\\
-N_{1}^{t}&N_{2}\end{pmatrix}=\begin{pmatrix}N_{0}&N_{1}\\\
N_{1}&N_{2}\end{pmatrix}$

We say that $M$ is of double skew format $F$ if there exists a matrix
$S\in\mathrm{GL}_{6}(\mathbb{C})$ such that $SMS^{t}$ is double skew and has
form $F$.

For future reference we recall the analogue of Laplace expansion for
Pfaffians.

######  Proposition 1.4.

Let $M$ be a skew-symmetric $2n\times 2n$-matrix (with entries in some
commutative ring), and $i\in\\{0,\dots 2n-1\\}$ fixed. Then

$\mathrm{Pf}(M)=\sum_{j\neq
i}(-1)^{i+j+1+\theta(i,j)}m_{ij}\mathrm{Pf}(M_{ij})$

where $\theta(i,j)=0$ if $i\leq j$ and $1$ otherwise.

###### Proof.

See [FP98, equation (D.1) p. 116]. ∎

##  2\. proof of the Theorem

---
Is the rank two
---
locus empty?
no yes Proposition 2.1
---
Manivel and
---
Mezzetti [MaMe05]
$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&*\\\ *&*&0&*&*&*\\\
*&*&*&0&0&0\\\ *&*&*&0&0&0\\\ *&*&*&0&0&0\\\ \end{smallmatrix}\right)$
---
hammer format
$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&*\\\ *&*&0&*&*&0\\\
*&*&*&0&*&0\\\ *&*&*&*&0&0\\\ *&*&0&0&0&0\\\ \end{smallmatrix}\right)$
---
arrow format
Proposition 2.2
---
Proposition 2.4
---

Figure 1. Flowchart for the proof of Theorem 2.5

In this section we prove our main Theorem. The structure of the proof is as
follows: If the matrix $M$ has no rank $2$ points the Theorem follows from
work of Manivel and Mezzetti [MaMe05]. If it does have a rank $2$ point we can
prove that $M$ has one of two special formats. For each of these formats we
prove the Theorem in a separate proposition. See Figure 1 for a flow chart of
this proof.

######  Proposition 2.1.

Let $M$ be a skew-symmetric $6\times 6$ matrix with vanishing Pfaffian and a
rank $2$ point. Then $M$ has one of the following formats

$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&*\\\ *&*&0&*&*&*\\\
*&*&*&0&0&0\\\ *&*&*&0&0&0\\\ *&*&*&0&0&0\\\
\end{smallmatrix}\right)\quad\left(\begin{smallmatrix}0&*&*&*&*&*\\\
*&0&*&*&*&*\\\ *&*&0&*&*&0\\\ *&*&*&0&*&0\\\ *&*&*&*&0&0\\\ *&*&0&0&0&0\\\
\end{smallmatrix}\right)$

Because of the shape of the zeros, we call the first format hammer format and
the second one arrow format.

###### Proof.

Since there exist a rank $2$ point we can assume that $M$ is of the form

$M=\left(\begin{smallmatrix}0&a&0&0&0&0\\\ a&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ \end{smallmatrix}\right)+N$

with $a\in V^{*}$ a non zero linear form and $a\not\in\langle
n_{01},\dots,n_{45}\rangle$. By Proposition 1.4, it follows that

$0=\mathrm{Pf}(M)=a\mathrm{Pf}(N_{01})+\mathrm{Pf}(N)$

and hence $\mathrm{Pf}(N_{01})=0$. From the Appendix we have that $N_{01}$ has
one of the following formats

$\left(\begin{smallmatrix}0&*&*&*\\\ *&0&0&0\\\ *&0&0&0\\\
*&0&0&0\end{smallmatrix}\right)\quad\left(\begin{smallmatrix}0&*&*&0\\\
*&0&*&0\\\ *&*&0&0\\\ 0&0&0&0\end{smallmatrix}\right).$

Since $N_{01}=M_{01}$ we obtain that $M$ is of format

$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&*\\\ *&*&0&*&*&*\\\
*&*&*&0&0&0\\\ *&*&*&0&0&0\\\ *&*&*&0&0&0\\\ \end{smallmatrix}\right)$

in the first case and of format

$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&*\\\ *&*&0&*&*&0\\\
*&*&*&0&*&0\\\ *&*&*&*&0&0\\\ *&*&0&0&0&0\\\ \end{smallmatrix}\right)$

in the second case. ∎

We start by analysing matrices that have hammer format:

######  Proposition 2.2.

Consider a skew $6\times 6$ matrix $M$ of linear form with hammer format

$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&*\\\ *&*&0&*&*&*\\\
*&*&*&0&0&0\\\ *&*&*&0&0&0\\\ *&*&*&0&0&0\\\ \end{smallmatrix}\right)$

and vanishing Pfaffian. Then $M$ has format $(a)$, $(b)$, $(c)$ or $(e)$ in
Table 1.

###### Proof.

In the situation of the Proposition we have two $3\times 3$ matrices of linear
forms $N$ and $N^{\prime}$ such that

$M=\begin{pmatrix}N^{\prime}&N\\\ -N^{t}&0\end{pmatrix}.$

We then have

$0=\mathrm{Pf}(M)=\det(N)$

From the Appendix we obtain that $N$ must have one of the following formats

$\left(\begin{smallmatrix}*&*&0\\\ *&*&0\\\
*&*&0\end{smallmatrix}\right)\quad\left(\begin{smallmatrix}*&*&*\\\ *&*&*\\\
0&0&0\end{smallmatrix}\right)\quad\left(\begin{smallmatrix}*&*&*\\\ *&0&0\\\
*&0&0\end{smallmatrix}\right)\quad\left(\begin{smallmatrix}0&*&*\\\ *&0&*\\\
*&*&0\end{smallmatrix}\right)$

the last one being skew. In these cases $M$ as format $(a)$, $(b)$, $(c)$ or
$(e)$ respectively. ∎

$\left(\begin{smallmatrix}0&*&*&*&*&\,\,a\,\,\\\ *&0&*&*&*&b\\\
*&*&0&y_{0}&y_{1}&0\\\ *&*&-y_{0}&0&y_{2}&0\\\ *&*&-y_{1}&-y_{2}&0&0\\\
-a&-b&0&0&0&0\\\ \end{smallmatrix}\right)$
---
$y_{0},y_{1},y_{2}$ linearly
---
independent?
no yes Proposition 2.2
---
$a,b$ linearly
---
independent?
no yes Lemma 2.3
---
$\mathrm{Pf}(M_{05})=bc$
---
$\mathrm{Pf}(M_{15})=ac$
$c\in\langle y_{0},y_{1},y_{2}\rangle$?
---
yes no $(d)$
---
$(f)$
---

Figure 2. Flowchart for the proof of Proposition 2.4

This leaves us to look at matrices that have arrow format. The proof is
divided into several cases, see Figure 2 for a flow chart of the proof. For
future reference we separate one case in the following Lemma:

######  Lemma 2.3.

Consider a skew $6\times 6$ matrix $M$ of linear forms with format

$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&0\\\ *&*&0&*&*&0\\\
*&*&*&0&*&0\\\ *&*&*&*&0&0\\\ *&0&0&0&0&0\\\ \end{smallmatrix}\right)$

and vanishing Pfaffian. Then $M$ has format $(a)$, $(b)$ or $(c)$ in Table 1.

###### Proof.

We can assume that $M$ is of the above form. If $m_{05}=0$ then $M$ has format
$(a)$ in Table 1

If $m_{05}\not=0$ then $\mathrm{Pf}(M_{05})$ must vanish (Pfaffian Laplace
expansion with respect to the last row). By Table 4 there exists a
$S\in\mathrm{GL}_{4}$ such that that $S^{t}M_{05}S$ is of the form

$\left(\begin{smallmatrix}0&*&*&*\\\ *&0&0&0\\\ *&0&0&0\\\
*&0&0&0\end{smallmatrix}\right)\quad\text{or}\quad\left(\begin{smallmatrix}0&*&*&0\\\
*&0&*&0\\\ *&*&0&0\\\ 0&0&0&0\end{smallmatrix}\right).$

Consequently, after operating with $\left(\begin{smallmatrix}1&0&0\\\ 0&S&0\\\
0&0&1\end{smallmatrix}\right)$ on $M$ we see that $M$ has format

$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&0\\\ *&*&0&0&0&0\\\
*&*&0&0&0&0\\\ *&*&0&0&0&0\\\ *&0&0&0&0&0\\\
\end{smallmatrix}\right)\quad\text{or}\quad\left(\begin{smallmatrix}0&*&*&*&*&*\\\
*&0&*&*&0&0\\\ *&*&0&*&0&0\\\ *&*&*&0&0&0\\\ *&0&0&0&0&0\\\ *&0&0&0&0&0\\\
\end{smallmatrix}\right)$

i.e. format $(b)$ or $(c)$ in Table 1. ∎

######  Proposition 2.4.

Consider a skew $6\times 6$ matrix $M$ of linear forms with arrow format

$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&*\\\ *&*&0&*&*&0\\\
*&*&*&0&*&0\\\ *&*&*&*&0&0\\\ *&*&0&0&0&0\\\ \end{smallmatrix}\right)$

and vanishing Pfaffian. Then $M$ has one of the formats in Table 1.

###### Proof.

After acting by an element in $\mathrm{GL}_{6}(\mathbb{C})$, we can assume
that $M$ is of the following form

$\left(\begin{smallmatrix}0&m_{01}&m_{02}&m_{03}&m_{04}&a\\\
-m_{01}&0&m_{12}&m_{13}&m_{14}&b\\\ -m_{02}&-m_{12}&0&y_{0}&y_{1}&0\\\
-m_{03}&-m_{13}&-y_{0}&0&y_{2}&0\\\ -m_{04}&-m_{14}&-y_{1}&-y_{2}&0&0\\\
-a&-b&0&0&0&0\end{smallmatrix}\right)$

Consider the subgroup $H\subset\mathrm{GL}_{6}$ consisting of matrices of the
form

$\begin{pmatrix}\mathrm{id}_{2}&S_{1}&S_{3}\\\ 0&S_{2}&S_{4}\\\
0&0&1\end{pmatrix}$

with $S_{1}$ a $2\times 3$ matrix, $S_{2}\in\mathrm{GL}_{3}$, $S_{3}$ a
$2\times 1$ matrix, and $S_{4}$ a $3\times 1$ matrix. $H$ operates on the
above $M$ without changing its form, i.e. $a,b$ and the zeros remain in their
places and only $S_{2}$ operates on $\left(\begin{smallmatrix}0&y_{0}&y_{1}\\\
-y_{0}&0&y_{2}\\\ -y_{1}&-y_{2}&0\\\ \end{smallmatrix}\right)$.

Notice that we can assume that $y_{0},y_{1},y_{2}$ are linearily independent.
If not there exists an $h\in H$ such that $hMh^{t}$ has the form

$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&*\\\ *&*&0&*&*&0\\\
*&*&*&0&0&0\\\ *&*&*&0&0&0\\\ *&*&0&0&0&0\\\ \end{smallmatrix}\right)$

and we are in the situation of Proposition 2.2.

Notice also that we can assume that $a$ and $b$ are linearly independent in
$V^{*}$. If not, we can assume that $M$ is of format

$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&0\\\ *&*&0&*&*&0\\\
*&*&*&0&0&0\\\ *&*&*&0&0&0\\\ *&0&0&0&0&0\\\ \end{smallmatrix}\right)$

and we are in the situation of Lemma 2.3

$\dim\langle a,b\rangle=2$: Here we have

$0=\mathrm{Pf}M=b\mathrm{Pf}(M_{15})-a\mathrm{Pf}(M_{05})$

and hence

$a\mathrm{Pf}(M_{05})=b\mathrm{Pf}(M_{15}).$

This implies that there exist a linear form $c\in V^{*}$, such that

$\mathrm{Pf}(M_{05})=bc\quad\text{and}\quad\mathrm{Pf}(M_{15})=ac.$

We now make a case distinction as to whether $c\in\langle
y_{0},y_{1},y_{2}\rangle$ (this includes the case $c=0$) or $c\not\in\langle
y_{0},y_{1},y_{2}\rangle$

$\dim\langle a,b\rangle=2,c\in\langle y_{0},y_{1},y_{2}\rangle$: Write

$c=\lambda_{0}y_{0}+\lambda_{1}y_{1}+\lambda_{2}y_{2}\ $

and consider

$S=\mathrm{id}_{\mathbb{C}^{6}}-\left(\begin{smallmatrix}0&0&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&\lambda_{2}\\\ 0&0&0&0&0&-\lambda_{1}\\\
0&0&0&0&0&\lambda_{0}\\\ 0&0&0&0&0&0\\\ \end{smallmatrix}\right)$

then

$N=SMS^{t}=M-\left(\begin{smallmatrix}0&0&a\lambda_{2}&-a\lambda_{1}&a\lambda_{0}&0\\\
0&0&b\lambda_{2}&-b\lambda_{1}&b\lambda_{0}&0\\\
-a\lambda_{2}&-b\lambda_{2}&0&0&0&0\\\ a\lambda_{1}&b\lambda_{1}&0&0&0&0\\\
-a\lambda_{0}&-b\lambda_{0}&0&0&0&0\\\ 0&0&0&0&0&0\\\
\end{smallmatrix}\right)$

Hence

$\mathrm{Pf}(N_{05})=\mathrm{Pf}(M_{05})-b(\lambda_{0}y_{0}+\lambda_{1}y_{1}+\lambda_{2}y_{2})=bc-
bc=0$

and similarly $\mathrm{Pf}(N_{15})=0$, compare [BB22M2].

Writing out the first Pfaffian we obtain

$0=n_{14}y_{0}-n_{13}y_{1}+n_{12}y_{2}=(y_{0},-y_{1},y_{2})\begin{pmatrix}n_{14}\\\
n_{13}\\\ n_{12}\end{pmatrix},$

i.e $(n_{14},n_{13},n_{14})^{t}$ is a linear syzygy of $(y_{0},-y_{1},y_{2})$.
The vector space of all linear syzygies of $(y_{0},-y_{1},y_{2})$ is generated
by the columns of

$\begin{pmatrix}0&{y}_{2}&{y}_{1}\\\ {-{y}_{2}}&0&{y}_{0}\\\
{-{y}_{1}}&{-{y}_{0}}&0\end{pmatrix}.$

Hence there exist constants $\alpha_{0},\alpha_{1},\alpha_{2}\in\mathbb{C}$
such that

$N=\left(\begin{smallmatrix}0&n_{01}&n_{02}&n_{03}&n_{04}&a\\\
-n_{01}&0&-\alpha_{0}y_{1}-\alpha_{1}y_{0}&-\alpha_{0}y_{2}+\alpha_{2}y_{0}&\alpha_{1}y_{2}+\alpha_{2}y_{1}&b\\\
-n_{02}&\alpha_{0}y_{1}+\alpha_{1}y_{0}&0&y_{0}&y_{1}&0\\\
-n_{03}&\alpha_{0}y_{2}-\alpha_{2}y_{0}&-y_{0}&0&y_{2}&0\\\
-n_{04}&-\alpha_{1}y_{2}-\alpha_{2}y_{1}&-y_{1}&-y_{2}&0&0\\\
-a&-b&0&0&0&0\end{smallmatrix}\right)$

We see that we can operate with an appropriate $S\in H$ to obtain

$\left(\begin{smallmatrix}0&n_{01}^{\prime}&n_{02}&n_{03}&n_{04}&a\\\
-n_{01}^{\prime}&0&0&0&0&b\\\ -n_{02}&0&0&y_{0}&y_{1}&0\\\
-n_{03}&0&-y_{0}&0&y_{2}&0\\\ -n_{04}&0&-y_{1}&-y_{2}&0&0\\\
-a&-b&0&0&0&0\end{smallmatrix}\right)$

using $\mathrm{Pf}(N_{15})=0$ in the same way we obtain

$\left(\begin{smallmatrix}0&n_{01}^{\prime}&0&0&0&a\\\
-n_{01}^{\prime}&0&0&0&0&b\\\ 0&0&0&y_{0}&y_{1}&0\\\ 0&0&-y_{0}&0&y_{2}&0\\\
0&0&-y_{1}&-y_{2}&0&0\\\ -a&-b&0&0&0&0\end{smallmatrix}\right).$

After rotating the last $4$ rows and columns cyclically, we see that $N$ and
hence $M$ is of format $(d)$.

$\dim\langle a,b\rangle=2,c\not\in\langle y_{0},y_{1},y_{2}\rangle$: Choosing
a basis of $V^{*}$ that contains $y_{0}$, $y_{1}$, $y_{2}$ and $c$, we let

$\overline{M}=\left(\begin{smallmatrix}0&\overline{m}_{01}&\overline{m}_{02}&\overline{m}_{03}&\overline{m}_{04}&\overline{a}\\\
-\overline{m}_{01}&0&\overline{m}_{12}&\overline{m}_{13}&\overline{m}_{14}&\overline{b}\\\
-\overline{m}_{02}&-\overline{m}_{12}&0&y_{0}&y_{1}&0\\\
-\overline{m}_{03}&-\overline{m}_{13}&-y_{0}&0&y_{2}&0\\\
-\overline{m}_{04}&-\overline{m}_{14}&-y_{1}&-y_{2}&0&0\\\
-\overline{a}&-\overline{b}&0&0&0&0\end{smallmatrix}\right)$

be the result of substituting $c=0$ into $M$.

We then have

$\mathrm{Pf}(\overline{M}_{05})=\overline{b}\overline{c}=0\quad\text{and}\quad\mathrm{Pf}(\overline{M}_{15})=\overline{a}\overline{c}=0$

As in the previous case we can find an $S\in H$ such that

$S\overline{M}S^{t}=\left(\begin{smallmatrix}0&*&0&0&0&\overline{a}\\\
*&0&0&0&0&\overline{b}\\\ 0&0&0&y_{0}&y_{1}&0\\\ 0&0&-y_{0}&0&y_{2}&0\\\
0&0&-y_{1}&-y_{2}&0&0\\\
-\overline{a}&-\overline{b}&0&0&0&0\end{smallmatrix}\right)$

Applying the same $S$ to $M$ gives

$M^{\prime}=SMS^{t}=\left(\begin{smallmatrix}0&m_{01}^{\prime}&\lambda_{02}c&\lambda_{03}c&\lambda_{04}c&a\\\
-m_{01}^{\prime}&0&\lambda_{12}c&\lambda_{13}c&\lambda_{14}c&b\\\
-\lambda_{02}c&-\lambda_{12}c&0&y_{0}&y_{1}&0\\\
-\lambda_{03}c&-\lambda_{13}c&-y_{0}&0&y_{2}&0\\\
-\lambda_{04}c&-\lambda_{14}c&-y_{1}&-y_{2}&0&0\\\
-a&-b&0&0&0&0\end{smallmatrix}\right)$

Computing the Pfaffian gives

$0=\mathrm{Pf}(M^{\prime})=-c\bigl{(}(\lambda_{14}{y}_{0}-\lambda_{13}{y}_{1}+\lambda_{12}{y}_{2})a-(\lambda_{04}{y}_{0}-\lambda_{03}{y}_{1}+\lambda_{02}{y}_{2})b\bigr{)}$

compare [BB22M2]. Since $c\not=0$

$(\lambda_{14}{y}_{0}-\lambda_{13}{y}_{1}+\lambda_{12}{y}_{2})a=(\lambda_{04}{y}_{0}-\lambda_{03}{y}_{1}+\lambda_{02}{y}_{2})b$

It follows that $a$ and $b$ are proportional to
$(\lambda_{04}{y}_{0}-\lambda_{03}{y}_{1}+\lambda_{02}{y}_{2})$ and
$(\lambda_{14}{y}_{0}-\lambda_{13}{y}_{1}+\lambda_{12}{y}_{2})$, i.e
$a,b\in\langle y_{0},y_{1},y_{2}\rangle$. We can therefore assume that after
another $H$-operation we have $y_{0}=-a$ and $y_{1}=-b$. The equation above is
then

$-\lambda_{14}a^{2}+\lambda_{13}ab+\lambda_{12}a{y}_{2}=-\lambda_{04}ab+\lambda_{03}b^{2}+\lambda_{02}b{y}_{2}$

and, by comparing coefficients we get

$\lambda_{14}=\lambda_{12}=\lambda_{03}=\lambda_{02}=0\text{ and
}\lambda_{13}=-\lambda_{04}=:-\lambda.$

This shows

$M^{\prime}=\left(\begin{smallmatrix}0&m_{01}^{\prime}&0&0&\lambda c&a\\\
-m_{01}^{\prime}&0&0&-\lambda c&0&b\\\ 0&0&0&-a&-b&0\\\ 0&\lambda
c&a&0&y_{2}&0\\\ -\lambda c&0&b&-y_{2}&0&0\\\
-a&-b&0&0&0&0\end{smallmatrix}\right)$

We see that $M^{\prime}$ and hence $M$ is of format $(f)$ in Table 1. ∎

Now we can prove:

######  Theorem 2.5.

Consider a skew $6\times 6$ matrix $M$ of linear forms with vanishing
Pfaffian. Then $M$ has one of the formats in Table 1.

###### Proof.

If $M$ has empty rank $2$ locus, then the Theorem follows from work of Manivel
and Mezzetti [MaMe05, Proposition 2, Theorem 4]. Indeed their cases $\ell_{s}$
and $\ell_{g}$ have format $(b)$, their case $\pi_{5}$ has format $(a)$, their
case $\pi_{t}$ has format $(b)$, their case $\pi_{p}$ has format $(c)$ and
their case $\pi_{g}$ has format $(d)$.

If $M$ does have a rank $2$ point, we know from Proposition 2.1 that $M$ has
one of the following formats

$\left(\begin{smallmatrix}0&*&*&*&*&*\\\ *&0&*&*&*&*\\\ *&*&0&*&*&*\\\
*&*&*&0&0&0\\\ *&*&*&0&0&0\\\ *&*&*&0&0&0\\\
\end{smallmatrix}\right)\quad\left(\begin{smallmatrix}0&*&*&*&*&*\\\
*&0&*&*&*&*\\\ *&*&0&*&*&0\\\ *&*&*&0&*&0\\\ *&*&*&*&0&0\\\ *&*&0&0&0&0\\\
\end{smallmatrix}\right)$

In the first case the claim follows from Proposition 2.2, in the second case
from Proposition 2.4. ∎

##  3\. Strictly semistable matrices

We consider the operation of $\mathrm{GL}_{6}(\mathbb{C})$ on the space of
skew symmetric $6\times 6$ matrices of linear forms given by

$M\mapsto SMS^{t}.$

Gaia Comaschi in [Co21, Thm. 3.1] has given a criterion in terms of formats to
check when a skew-symmetric $6\times 6$ matrix of linear forms is not stable
or not semistable with respect to this action:

######  Theorem 3.1.

Let $M$ be a skew-symmetric $6\times 6$ matrix of linear forms. Then $M$ is
not stable, therefore strictly semistable or unstable, if and only if $M$ has
one of the following skew formats:

$(\mathrm{NS}_{1}):\left(\begin{smallmatrix}0&0&0&0&0&*\\\ 0&0&*&*&*&*\\\
0&*&0&*&*&*\\\ 0&*&*&0&*&*\\\ 0&*&*&*&0&*\\\
*&*&*&*&*&0\end{smallmatrix}\right)\quad(\mathrm{NS}_{2}):\left(\begin{smallmatrix}0&0&0&0&*&*\\\
0&0&0&0&*&*\\\ 0&0&0&*&*&*\\\ 0&0&*&0&*&*\\\ *&*&*&*&0&*\\\
*&*&*&*&*&0\end{smallmatrix}\right)$

$(\mathrm{NS}_{3}):\left(\begin{smallmatrix}0&0&0&*&*&*\\\ 0&0&0&*&*&*\\\
0&0&0&*&*&*\\\ *&*&*&0&*&*\\\ *&*&*&*&0&*\\\
*&*&*&*&*&0\end{smallmatrix}\right)$

Furthermore, $M$ is not semistable, or equivalently, unstable, if and only if
it has one of the following skew formats:

$(\mathrm{NSS}_{1}):\left(\begin{smallmatrix}0&0&0&0&0&0\\\ 0&0&*&*&*&*\\\
0&*&0&*&*&*\\\ 0&*&*&0&*&*\\\ 0&*&*&*&0&*\\\
0&*&*&*&*&0\end{smallmatrix}\right)\quad(\mathrm{NSS}_{2}):\left(\begin{smallmatrix}0&0&0&0&0&*\\\
0&0&0&0&0&*\\\ 0&0&0&*&*&*\\\ 0&0&*&0&*&*\\\ 0&0&*&*&0&*\\\
*&*&*&*&*&0\end{smallmatrix}\right)$

$(\mathrm{NSS}_{3}):\left(\begin{smallmatrix}0&0&0&0&*&*\\\ 0&0&0&0&*&*\\\
0&0&0&0&*&*\\\ 0&0&0&0&*&*\\\ *&*&*&*&0&*\\\
*&*&*&*&*&0\end{smallmatrix}\right)$

Notice that Comaschi uses the convention of collecting the zeros in the upper
left corner, while we (following Manivel and Mezzetti [MaMe05]) collect the
zeros in the lower right corner. To convert between these conventions we say
that the $r\times r$ matrix

$M^{\mathrm{rev}}=(m_{r-i-1,r-j-1})_{0\leq i,j<r}$

is obtained from $M=(m_{i,j})_{0\leq i,j<r}$ by reversing rows and columns. We
also call $M^{\mathrm{rev}}$ the reverse matrix. Notice that if $M$ has format
$F$ it also has the format described by the reverse of $F$.

We now assume in addition that the Pfaffian of $M$ vanishes. If $M$ is stable
it has to have format $(d)$ or $(f)$ in Table 1 since $(a)$, $(b)$, $(c)$ are
unstable by the above Theorem and matrices of format $(e)$ are not stable.
Comaschi gives normal forms for the stable cases. At the end of this section
we reprove her results using our methods. Before that we extend Comaschi’s
classification to the strictly semistable case. We start by narrowing down the
possible format of a strictly semistable matrix $M$:

$M$ strictly semistable,
---
$\mathrm{Pf}(M)=0$
$\mathrm{NS}_{1}$
---
$\left(\begin{smallmatrix}0&0&0&0&0&*\\\ 0&0&*&*&*&*\\\ 0&*&0&*&*&*\\\
0&*&*&0&*&*\\\ 0&*&*&*&0&*\\\ *&*&*&*&*&0\end{smallmatrix}\right)$
$\mathrm{NS}_{2}$
---
$\left(\begin{smallmatrix}0&0&0&0&*&*\\\ 0&0&0&0&*&*\\\ 0&0&0&*&*&*\\\
0&0&*&0&*&*\\\ *&*&*&*&0&*\\\ *&*&*&*&*&0\end{smallmatrix}\right)$
$\mathrm{NS}_{3}$
---
$\left(\begin{smallmatrix}0&0&0&*&*&*\\\ 0&0&0&*&*&*\\\ 0&0&0&*&*&*\\\
*&*&*&0&*&*\\\ *&*&*&*&0&*\\\ *&*&*&*&*&0\end{smallmatrix}\right)$
Lemma 2.3
---
Proposition 2.2
---
$(a)$, $(b)$ or $(c)$
---
$(a)$ or $(c)$
---
$(a)$, $(b)$ or $(c)$
---
$(e)$
---
unstable
---
Proposition 3.3
---

Figure 3. Flowchart for the classification of strictly semistable matrices $M$
with $\mathrm{Pf}(M)=0$

######  Proposition 3.2.

Let $M$ be a strictly semistable, skew symmetric $6\times 6$ matrix of linear
forms with vanishing Pfaffian. Then $M$ has double skew format

$\left(\begin{smallmatrix}0&0&0&0&*&*\\\ 0&0&0&*&0&*\\\ 0&0&0&*&*&0\\\
0&*&*&0&*&*\\\ *&0&*&*&0&*\\\ *&*&0&*&*&0\\\ \end{smallmatrix}\right).$

###### Proof.

By Theorem 3.1 we know that $M$ has format $NS_{1}$, $NS_{2}$ or $NS_{3}$. By
assumption $\mathrm{Pf}(M)=0$. We treat each case in turn:

$NS_{1}$: If $M$ has format $NS_{1}$ we can reverse rows and columns and apply
Lemma 2.3. It follows that $M$ has format $(a)$, $(b)$ or $(c)$ in Table 1.
But then format $(a)$ is the reverse of $NSS_{1}$, format $(b)$ is reverse of
$NSS_{3}$ and format $(c)$ is the reverse $NSS_{2}$. Therefore $M$ must be
unstable.

$NS_{2}$: We can assume, that $M$ is of the form

$M=\begin{pmatrix}0&0&C\\\ 0&D&B\\\ -C^{t}&-B^{t}&A\end{pmatrix}$

with $2\times 2$ matrices $A,B,C,D$ of which $A$ and $D$ are skew. We then
have

$0=\mathrm{Pf}(M)=\det(C)\cdot\mathrm{Pf}(D)$

If $\mathrm{Pf}(D)=0$ then $D=0$ since $D$ is a $2\times 2$ matrix. But then
$M$ has format $NSS_{3}$ and is therefore unstable. This leaves the case
$\det(C)=0$. By reversing the formats in Table 4 we know that the matrix $C$
has format

$\left(\begin{smallmatrix}0&*\\\
0&*\end{smallmatrix}\right)\quad\text{or}\quad\left(\begin{smallmatrix}0&0\\\
*&*\end{smallmatrix}\right).$

In the first case $M$ has format $NSS_{2}$, in the second case $NSS_{1}$.
Again $M$ must be unstable.

$NS_{3}$: In this case $M$ has hammer format and $\mathrm{Pf}(M)=0$, so we are
(after reversing rows and columns of $M$) in the situation of Proposition 2.2.
It follows that $M$ has format $(a)$, $(b)$, $(c)$ or $(e)$ in Table 1. $(a)$,
$(b)$ and $(c)$ are again unstable, so that $M$ must have format double skew
format $(e)$. Reversing format $(e)$ we obtain the claim. ∎

######  Proposition 3.3.

Let $M$ be a strictly semistable, skew-symmetric $6\times 6$ matrix of linear
forms with vanishing Pfaffian. Let $r=\dim\mathbb{P}\langle
m_{01},\dots,m_{45}\rangle$ the dimension of the projective space spanned by
the entries of $M$. Then $r\in\\{2,3,4,5\\}$ and there exist a matrix
$S\in\mathrm{GL}_{6}(\mathbb{C})$ and linear forms $x_{0},\dots,x_{5}$ such
that

$S^{t}MS=\left(\begin{smallmatrix}0&0&0&0&x_{0}&x_{1}\\\
0&0&0&-x_{0}&0&x_{2}\\\ 0&0&0&-x_{1}&-x_{2}&0\\\
0&x_{0}&x_{1}&0&x_{3}&x_{4}\\\ -x_{0}&0&x_{2}&-x_{3}&0&x_{5}\\\
-x_{1}&x_{2}&0&-x_{4}&-x_{5}&0\\\ \end{smallmatrix}\right)$

with $x_{0},\dots,x_{r}$ linearly independent and $x_{i}=0$ for $i>r$.

###### Proof.

We write $M$ as

$M=\begin{pmatrix}0&B\\\ -B^{t}&A\end{pmatrix}$

with $A,B$ skew-symmetric $3\times 3$ matrices of linear forms. If the entries
of $B$ are linearly dependent, $B$ has skew format

$\left(\begin{smallmatrix}0&0&*\\\ 0&0&*\\\ *&*&0\\\
\end{smallmatrix}\right).$

and hence $M$ has format $NSS_{2}$. So the entries of $B$ must be linearly
independent.

We can now write $A=A^{\prime}+A^{\prime\prime}$ such that

  1. a)

$A^{\prime}$ and $A^{\prime\prime}$ are skew-symmetric $3\times 3$ matrices of
linear forms.

  2. b)

$\langle A^{\prime}\rangle\subset\langle B\rangle$, i.e the entries of
$A^{\prime}$ are in the span of the entries of $B$.

  3. c)

$\langle A^{\prime\prime}\rangle\cap\langle B\rangle=\\{0\\}$, i.e. the
entries of $A^{\prime\prime}$ are independent of the entries of $B$.

Because of condition $b)$ there exists a not necessarily invertible $3\times
3$ matrix $T$ such that

$TB-B^{t}T^{t}=A^{\prime}.$

Setting

$S=\begin{pmatrix}\mathrm{id}&0\\\ -T&\mathrm{id}\end{pmatrix}$

we have

$SMS^{t}=\begin{pmatrix}0&B\\\
-B^{t}&A-TB+B^{t}T^{t}\end{pmatrix}=\begin{pmatrix}0&B\\\
-B^{t}&A^{\prime\prime}\end{pmatrix}$

We can therefore choose a basis of $x_{3},\dots,x_{r}$ of $\langle
A^{\prime\prime}\rangle$ and assume that after another
$\mathrm{GL}_{6}(\mathbb{C})$ operation

$A^{\prime\prime}=\left(\begin{smallmatrix}0&x_{3}&x_{4}\\\ -x_{3}&0&x_{5}\\\
-x_{4}&-x_{5}&0\\\ \end{smallmatrix}\right).$

with $x_{i}=0$ for $i>r$. We now set $x_{0}=b_{01},x_{1}=b_{02}$ and
$x_{2}=b_{12}$, i.e

$B=\left(\begin{smallmatrix}0&x_{0}&x_{1}\\\ -x_{0}&0&x_{2}\\\
-x_{1}&-x_{2}&0\\\ \end{smallmatrix}\right).$

Since the entries of $B$ are linearly independent and $\langle
B\rangle\cap\langle A^{\prime\prime}\rangle=\\{0\\}$ we have that
$x_{0},\dots,x_{r}$ are also linearly independent and $S^{t}MS$ has the
claimed form. ∎

$M$ stable,
---
$\mathrm{Pf}(M)=0$
$(a)$
---
$(b)$
---
$(c)$
---
$(d)$
---
$(e)$
---
$(f)$
---
$NSS_{1}$
---
$NSS_{3}$
---
$NSS_{2}$
---
Proposition 3.4
---
$NS_{3}$
---
Proposition 3.5
---

Figure 4. Flowchart for the classification of stable matrices $M$ with
$\mathrm{Pf}(M)=0$

######  Proposition 3.4.

If $M$ is a stable skew-symmetric $6\times 6$ matrix of linear forms of format
$(d)$ then after an $\mathrm{GL}_{6}(\mathbb{C})$ operation we have

$M=\left(\begin{smallmatrix}0&{l}_{0}&{l}_{1}\\\ {-{l}_{0}}&0&{l}_{2}\\\
{-{l}_{1}}&{-{l}_{2}}&0\\\ &&&0&{m}_{0}&{m}_{1}\\\ &&&{-{m}_{0}}&0&{m}_{2}\\\
&&&{-{m}_{1}}&{-{m}_{2}}&0\end{smallmatrix}\right)$

with $\dim\langle l_{0},l_{1},l_{2}\rangle=\dim\langle
m_{0},m_{1},m_{2}\rangle=3$.

###### Proof.

If $M$ is of format $(d)$ we have

$M=\left(\begin{smallmatrix}0&{l}_{0}&{l}_{1}\\\ {-{l}_{0}}&0&{l}_{2}\\\
{-{l}_{1}}&{-{l}_{2}}&0\\\ &&&0&{m}_{0}&{m}_{1}\\\ &&&{-{m}_{0}}&0&{m}_{2}\\\
&&&{-{m}_{1}}&{-{m}_{2}}&0\end{smallmatrix}\right)$

with as yet no information about linear independence of the entries. If
$\dim\langle l_{0},l_{1},l_{2}\rangle<3$ we have, can obtain, after a
$\mathrm{GL}_{6}(\mathbb{C})$-operation

$M=\left(\begin{smallmatrix}0&{l}_{0}&{l}_{1}\\\ {-{l}_{0}}&0&0\\\
{-{l}_{1}}&0&0\\\ &&&0&{m}_{0}&{m}_{1}\\\ &&&{-{m}_{0}}&0&{m}_{2}\\\
&&&{-{m}_{1}}&{-{m}_{2}}&0\end{smallmatrix}\right)$

which is unstable because it has format $NSS_{2}$. ∎

######  Proposition 3.5.

If $M$ is a stable skew-symmetric $6\times 6$ matrix of linear forms of format
$(f)$ that is not of format $(d)$, then after a
$\mathrm{GL}_{6}(\mathbb{C})$-operation we have

$M=\left(\begin{smallmatrix}&{l}_{3}&&0&{l}_{0}&{l}_{1}\\\
{-{l}_{3}}&&&{-{l}_{0}}&0&{l}_{2}\\\ &&&{-{l}_{1}}&{-{l}_{2}}&0\\\
0&{l}_{0}&{l}_{1}&&{l}_{4}&\\\ {-{l}_{0}}&0&{l}_{2}&{-{l}_{4}}&&\\\
{-{l}_{1}}&{-{l}_{2}}&0&&&\end{smallmatrix}\right)$

with $l_{0},\dots,l_{4}$ linearly independent.

###### Proof.

Since $M$ is of format $(f)$ we can assume

$M=\left(\begin{smallmatrix}&{l}_{3}&&0&{l}_{0}&m\\\
{-{l}_{3}}&&&{-{l}_{0}}&0&n\\\ &&&{-m}&{-n}&0\\\ 0&{l}_{0}&m&&{l}_{4}&\\\
{-{l}_{0}}&0&n&{-{l}_{4}}&&\\\ {-m}&{-n}&0&&&\end{smallmatrix}\right)$

If $m$ and $n$ are linearly dependent we are in the situation of Lemma 2.3 and
$M$ is unstable. We can therefore assume that $m$ and $n$ are linearly
independent.

Applying the permutation $(0,1,2,3,4,5)\mapsto(0,3,1,4,5,2)$ to rows and
columns we can write $M$ as a $3\times 3$ matrix of symmetric $2\times 2$
matrices

$M=\begin{pmatrix}0&A&m\cdot\mathrm{id}\\\ -A&0&n\cdot\mathrm{id}\\\
-m\cdot\mathrm{id}&-n\cdot\mathrm{id}&0\end{pmatrix}$

with $A=\left(\begin{smallmatrix}l_{3}&l_{0}\\\
l_{0}&l_{4}\end{smallmatrix}\right)$ a _symmetric_ $2\times 2$-matrix
[BB22M2]. We now write

$A=A^{\prime}+mA^{\prime\prime}+nA^{\prime\prime\prime}$

with $A^{\prime\prime}$ and $A^{\prime\prime\prime}$ symmetric $2\times 2$
matrices over $\mathbb{C}$ and $A^{\prime}$ a symmetric $2\times 2$ matrix of
linear forms with $\langle A^{\prime}\rangle\cap\langle m,n\rangle=\\{0\\}$.
With

$S=\begin{pmatrix}\mathrm{id}&0&A^{\prime\prime\prime}\\\
0&\mathrm{id}&-A^{\prime\prime}\\\ 0&0&\mathrm{id}\end{pmatrix}$

we have

$\displaystyle M^{\prime}=SMS^{t}$
$\displaystyle=\begin{pmatrix}0&A-mA^{\prime\prime}-nA^{\prime\prime\prime}&m\cdot\mathrm{id}\\\
-(A-mA^{\prime\prime}-nA^{\prime\prime\prime})&0&n\cdot id\\\
-m\cdot\mathrm{id}&-n\cdot\mathrm{id}&0\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&A^{\prime}&m\cdot\mathrm{id}\\\
-A^{\prime}&0&n\cdot\mathrm{id}\\\
-m\cdot\mathrm{id}&-n\cdot\mathrm{id}&0\end{pmatrix}$

compare [BB22M2]. Assume now to the contrary

$5>\dim\langle M\rangle=\dim\langle m,n\rangle+\dim\langle
A^{\prime}\rangle=2+\dim\langle A^{\prime}\rangle$

and therefore $\dim\langle A^{\prime}\rangle<3$. We then have a matrix
$T\in\mathrm{GL}_{2}(\mathbb{C})$ such that $TA^{\prime}T^{t}$ is

$\left(\begin{smallmatrix}a&b\\\
b&0\end{smallmatrix}\right)\quad\text{or}\quad\left(\begin{smallmatrix}a&0\\\
0&b\end{smallmatrix}\right)$

with $a,b$ possibly zero linear forms. Operating with

$\begin{pmatrix}T^{t}&0&0\\\ 0&T^{t}&0\\\ 0&0&T^{-1}\end{pmatrix}$

we see that we can assume that $A^{\prime}$ is
$\left(\begin{smallmatrix}a&b\\\ b&0\end{smallmatrix}\right)$ or
$\left(\begin{smallmatrix}a&0\\\ 0&b\end{smallmatrix}\right)$. We apply the
reverse of the permutation above. In the first case we obtain

$\left(\begin{smallmatrix}&{a}&&0&b&m\\\ -{a}&&&{-b}&0&n\\\ &&&{-m}&{-n}&0\\\
0&b&m&&0&\\\ {-b}&0&n&&&\\\ {-m}&{-n}&&&&\end{smallmatrix}\right)$

and $M$ is not stable by $NS_{3}$. In the second case we obtain

$\left(\begin{smallmatrix}0&{a}&0&0&0&m\\\ -{a}&0&0&0&0&n\\\
0&0&0&{-m}&{-n}&0\\\ 0&0&m&0&b&0\\\ 0&0&n&-b&0&0\\\
{-m}&{-n}&0&0&0&0\end{smallmatrix}\right)$

rearranging the rows and columns again, we get

$\left(\begin{smallmatrix}0&{a}&m&0&0&0\\\ -{a}&0&n&0&0&0\\\ -m&-n&0&0&0&0\\\
0&0&0&0&{-m}&{-n}\\\ 0&0&0&m&0&b\\\ 0&0&0&n&-b&0\\\ \end{smallmatrix}\right)$

and $M$ has format $(d)$, contrary to our assumption. We therefore must have
$\dim\langle M\rangle=5$ as claimed. ∎

##  4\. $\mathbb{P}^{4}$

For applications semistable $6\times 6$ matrices of linear forms over
$\mathbb{P}^{4}$ with vanishing Pfaffian seem to be particularly interesting.
We collect here the normal forms of such matrices for future reference.

| $M$ | $Y^{\mathrm{top}}$ | $d_{4}$ | stability
---|---|---|---|---
(a) | $\left(\begin{smallmatrix}&{l}_{3}&&0&{l}_{0}&{l}_{1}\\\ {-{l}_{3}}&&&{-{l}_{0}}&0&{l}_{2}\\\ &&&{-{l}_{1}}&{-{l}_{2}}&0\\\ 0&{l}_{0}&{l}_{1}&&{l}_{4}&\\\ {-{l}_{0}}&0&{l}_{2}&{-{l}_{4}}&&\\\ {-{l}_{1}}&{-{l}_{2}}&0&&&\end{smallmatrix}\right)$ | a smooth conic | 10 | stable
(b) | $\left(\begin{smallmatrix}0&{l}_{0}&{l}_{1}\\\ {-{l}_{0}}&0&{l}_{2}\\\ {-{l}_{1}}&{-{l}_{2}}&0\\\ &&&0&{l}_{2}&{l}_{3}\\\ &&&{-{l}_{2}}&0&{l}_{4}\\\ &&&{-{l}_{3}}&{-{l}_{4}}&0\end{smallmatrix}\right)$ | two skew lines | 9 | stable
(c) | $\left(\begin{smallmatrix}0&{l}_{0}&{l}_{1}\\\ {-{l}_{0}}&0&{l}_{2}\\\ {-{l}_{1}}&{-{l}_{2}}&0\\\ &&&0&{l}_{1}&{l}_{2}\\\ &&&{-{l}_{1}}&0&{l}_{3}\\\ &&&{-{l}_{2}}&{-{l}_{3}}&0\end{smallmatrix}\right)$ |  | two distinct
---
intersecting lines
8 | stable
(d) | $\left(\begin{smallmatrix}&&&0&l_{0}&l_{1}\\\ &&&-l_{0}&0&l_{2}\\\ &&&-l_{1}&-l_{2}&0\\\ 0&l_{0}&l_{1}&&l_{3}&l_{4}\\\ -l_{0}&0&l_{2}&-l_{3}&&\\\ -l_{1}&-l_{2}&0&-l_{4}&&\end{smallmatrix}\right)$ |  | a double line lying on
---
a smooth quadric surface
9 |  | strictly semistable,
---
but not polystable
(e) | $\left(\begin{smallmatrix}&&&0&l_{0}&l_{1}\\\ &&&-l_{0}&0&l_{2}\\\ &&&-l_{1}&-l_{2}&0\\\ 0&l_{0}&l_{1}&&l_{3}&\\\ -l_{0}&0&l_{2}&-l_{3}&&\\\ -l_{1}&-l_{2}&0&&&\end{smallmatrix}\right)$ | a plane double line | 8 |  | strictly semistable,
---
but not polystable
(f) | $\left(\begin{smallmatrix}&&&0&l_{0}&l_{1}\\\ &&&-l_{0}&0&l_{2}\\\ &&&-l_{1}&-l_{2}&0\\\ 0&l_{0}&l_{1}&&&\\\ -l_{0}&0&l_{2}&&&\\\ -l_{1}&-l_{2}&0&&&\end{smallmatrix}\right)$ |  | a line together with its
---
full first order
infinitesimal
neighbourhood
6 | polystable
Table 2. Semistable matrices $M$ with vanishing Pfaffian over $\mathbb{P}^{4}$

######  Theorem 4.1.

Let $M$ be a semistable skew-symmetric $6\times 6$ matrix with vanishing
Pfaffian over $\mathbb{P}^{4}$. Then there exists linearly independent linear
forms $l_{0},\dots,l_{4}$ and a matrix $S\in\mathrm{GL}_{6}(\mathbb{C})$ such
that $SMS^{t}$ is one of the matrices in Table 2.

###### Proof.

If $M$ is stable and $\mathrm{Pf}(M)=0$ it must be of format $(d)$ or $(f)$ in
Table 1. If it is of format $(d)$ we obtain the normal form $(b)$ and $(c)$ of
Table 2 from Proposition 3.4. If it is of format $(f)$ but not of format $(d)$
in Table 1 we obtain the normal form $(a)$ of Table 2 from Proposition 3.5.

If $M$ is strictly semistable we can apply our Proposition 3.3: The case $r=5$
cannot occur since we are in $\mathbb{P}^{4}$; $r=4$ gives case $(d)$, $r=3$
is case $(e)$ and $r=2$ is case $(f)$ in Table 2. ∎

######  Proposition 4.2.

Let $M$ be a semi-stable skew-symmetric $6\times 6$ matrix with vanishing
Pfaffian over $\mathbb{P}^{4}$. Than $M$ is congruent to at most one of the
normal forms in Table 2,.

###### Proof.

Let $Y$ be the rank $2$ locus of $M$ with the scheme structure given by the
$4\times 4$ Pfaffians and $Y^{\mathrm{top}}$ the top dimensional component of
$Y$. We compute $Y^{\mathrm{top}}$ for all matrices in Table 2 using Macaulay2
see [BB22M2]. The results are recorded in the third column of Table 2. Since
$Y^{\mathrm{top}}$ is different in each case, the claim follows. ∎

To determine the stability properties of the normal forms in Table 2 we
provide two methods to prove that a given matrix is not of a particular
format. The first method is by looking at the dimension of the space of
quadrics spanned by the $4\times 4$ Pfaffians $M$ which we denote by
$d_{4}(M):=\dim\mathrm{Pf}_{4}(M)$. Notice that this is invariant under the
$\mathrm{GL}_{6}(\mathbb{C})$-action.

| Format | $d_{4}$ | generalized rows
---|---|---|---
$\mathrm{NS}_{1}$ | $\left(\begin{smallmatrix}0&0&0&0&0&*\\\ 0&0&*&*&*&*\\\ 0&*&0&*&*&*\\\ 0&*&*&0&*&*\\\ 0&*&*&*&0&*\\\ *&*&*&*&*&0\end{smallmatrix}\right)$ | $\leq 11$ | $\mathbb{P}^{0}\subset Z_{1}$
$\mathrm{NS}_{2}$ | $\left(\begin{smallmatrix}0&0&0&0&*&*\\\ 0&0&0&0&*&*\\\ 0&0&0&*&*&*\\\ 0&0&*&0&*&*\\\ *&*&*&*&0&*\\\ *&*&*&*&*&0\end{smallmatrix}\right)$ | $\leq 10$ | $\mathbb{P}^{1}\subset Z_{2}$
$\mathrm{NS}_{3}$ | $\left(\begin{smallmatrix}0&0&0&*&*&*\\\ 0&0&0&*&*&*\\\ 0&0&0&*&*&*\\\ *&*&*&0&*&*\\\ *&*&*&*&0&*\\\ *&*&*&*&*&0\end{smallmatrix}\right)$ | $\leq 12$ | $\mathbb{P}^{2}\subset Z_{3}$
$\mathrm{NSS}_{1}$ | $\left(\begin{smallmatrix}0&0&0&0&0&0\\\ 0&0&*&*&*&*\\\ 0&*&0&*&*&*\\\ 0&*&*&0&*&*\\\ 0&*&*&*&0&*\\\ 0&*&*&*&*&0\end{smallmatrix}\right)$ | $\leq 5$ | $\mathbb{P}^{0}\subset Z_{0}$
$\mathrm{NSS}_{2}$ | $\left(\begin{smallmatrix}0&0&0&0&0&*\\\ 0&0&0&0&0&*\\\ 0&0&0&*&*&*\\\ 0&0&*&0&*&*\\\ 0&0&*&*&0&*\\\ *&*&*&*&*&0\end{smallmatrix}\right)$ | $\leq 7$ | $\mathbb{P}^{1}\subset Z_{1}$
$\mathrm{NSS}_{3}$ | $\left(\begin{smallmatrix}0&0&0&0&*&*\\\ 0&0&0&0&*&*\\\ 0&0&0&0&*&*\\\ 0&0&0&0&*&*\\\ *&*&*&*&0&*\\\ *&*&*&*&*&0\end{smallmatrix}\right)$ | $\leq 6$ | $\mathbb{P}^{3}\subset Z_{2}$
Table 3. Invariants of not stable and not semi-stable formats

######  Proposition 4.3.

For the formats $\mathrm{NS_{i}}$ and $\mathrm{NSS_{i}}$ the invariant $d_{4}$
is as listed in Table 3. In particular if $d_{4}(M)\geq 8$ then $M$ is semi-
stable and if $d_{4}(M)\geq 13$ then $M$ is stable.

###### Proof.

We compute the invariant $d_{4}$ for the generic matrices of each format, i.e.
those with linearly independent linear forms at the position of the $*$’s. For
arbitrary entries the dimension of the space of quadrics spanned by the
$4\times 4$ Pfaffians can only be smaller than the dimension in the generic
case.

For example in the case $\mathrm{NSS_{3}}$ the $4\times 4$ Pfaffians of $M$
are just the $2\times 2$ minors of the $2\times 4$ matrix at the position of
the $*$’s. This gives at most $6$ independent $4\times 4$ Pfaffians. In the
other cases we compute $d_{4}$ in the generic case with Macaulay2 [BB22M2].

If $d_{4}(M)$ is at least $8$ it cannot have one of the formats
$\mathrm{NSS_{i}}$ since $d_{4}(M)$ is too large. Therefore it cannot be not
semi-stable. Similarly if $d_{4}(M)$ is at least $13$ then $M$ cannot have
formats $\mathrm{NS_{i}}$. Therefore in this case $M$ cannot be not stable. ∎

######  Corollary 4.4.

The matrices of type $(a)$ to $(e)$ in Table 2 are semi-stable.

###### Proof.

We compute $d_{4}(M)$ for the matrices of type $(a)$ to $(f)$ in Table 2, see
[BB22M2]. The results are listed in the fourth column of Table 2. We then
apply Proposition 4.3 in the cases $(a)$ to $(e)$. ∎

In some cases the invariant $d_{4}$ does not give enough information. For a
finer invariant we introduce the concept of generalized rows and their rank
which we learned from David Eisenbud.

######  Definition 4.5.

Let $V,W$ be $\mathbb{C}$ vector spaces. After choosing bases of $V$ and $W$
we have a $1:1$ correspondence between $\dim W\times\dim W$ matrices of linear
forms over $\mathbb{P}(V)$ and elements of

$\mathrm{Hom}(V,W^{*}\otimes W^{*}).$

In this situation we call $\mathbb{P}(W)$ the space of generalized rows.
Considering the natural isomorphisms

$\mathrm{Hom}(V,W^{*}\otimes W^{*})\cong W^{*}\otimes W^{*}\otimes
V^{*}\cong\mathrm{Hom}(W,V^{*}\otimes W^{*})$

we see that (after choosing bases for $V$ and $W$) there is a $1:1$
correspondence between $\dim W\times\dim W$ matrices of linear forms over
$\mathbb{P}(V)$ and $\dim V\times\dim W$ matrices of linear forms over
$\mathbb{P}(W)$. We denote this correspondence by

$M\mapsto\widehat{M}.$

and call $\widehat{M}$ the flipped matrix of $M$.

We define the rank of a generalized row $[w]\in\mathbb{P}(W)$ as the rank of
$\widehat{M}(w)$. Indeed if $[w]$ corresponds to a row of $M$, the rank of
$[w]$ will be the dimension of the vector space spanned by the linear forms in
this row. We also let $Z_{r}\subset\mathbb{P}(W)$ be the space of generalized
rows with rank at most $r$ with the scheme structure given by the
$(r+1)\times(r+1)$ minors of $\widehat{M}$.

For $S\in\mathrm{GL}(W)$ the operation $M\mapsto SMS^{t}$ corresponds to a
change of basis in $W$. Therefore the schemes $Z_{r}(M)$ and $Z_{r}(SMS^{t})$
in $\mathbb{P}(W)$ differ only by a linear transformation of $\mathbb{P}(W)$.

######  Example 4.6.

Consider matrices of the form $\mathrm{NSS_{3}}$. The first four rows and all
linear combinations of them have the first $4$ entries equal to zero.
Therefore the space of rank at most $2$ generalized rows $Z_{2}$ contains a
$\mathbb{P}^{3}$. In the same way the other formats of Table 3 have large
linear spaces of low rank generalized rows. We have collected this information
in Table 3.

For a given matrix $M$ the flipped matrix can be easily computed:

######  Proposition 4.7.

Let $V,W$ be $\mathbb{C}$-vector spaces with bases $l_{0},\dots l_{m}$ of
$V^{*}$ and $y_{0},\dots,y_{n}$ of $W^{*}$. Let furthermore $M$ be a $\dim
W\times\dim W$ matrix of linear forms over $\mathbb{P}(V)$ written in the
above bases and $\widehat{M}$ be the flipped matrix of $M$. Then

$\widehat{M}_{i}=\frac{\partial}{\partial
l_{i}}\bigl{(}(y_{0},\dots,y_{n})M\bigr{)},\quad i=0,\dots,m$

where $\widehat{M}_{i}$ is the $i$-th row of $\widehat{M}$. Notice that
$\widehat{M}$ is a $\dim V\times\dim W$ matrix.

###### Proof.

It suffices to check for matrices $M$ with a single nonzero entry equal to one
of the $l_{i}$, i.e. monomials in $W^{*}\otimes W^{*}\otimes V^{*}$. ∎

######  Example 4.8.

Consider the matrix

$M=\left(\begin{smallmatrix}&&&0&l_{0}&l_{1}\\\ &&&-l_{0}&0&l_{2}\\\
&&&-l_{1}&-l_{2}&0\\\ 0&l_{0}&l_{1}&&&\\\ -l_{0}&0&l_{2}&&&\\\
-l_{1}&-l_{2}&0&&&\end{smallmatrix}\right).$

of type $(f)$. We compute the flipped matrix as

$\widehat{M}=\left(\frac{\partial}{\partial
l_{i}}\bigl{(}(y_{0},\dots,y_{5})M_{f}\bigr{)}\right)=\left(\begin{smallmatrix}{-{y}_{4}}&{y}_{3}&0&{-{y}_{1}}&{y}_{0}&0\\\
{-{y}_{5}}&0&{y}_{3}&{-{y}_{2}}&0&{y}_{0}\\\
0&{-{y}_{5}}&{y}_{4}&0&{-{y}_{2}}&{y}_{1}\end{smallmatrix}\right)$

It can be checked by a computer algebra calculation [BB22M2] that the rank
$\leq 1$ locus $Z_{1}(M)$ defined by the two by two minors of $\widehat{M}$ is
empty. Therefore $M$ cannot be of format $\mathrm{NSS_{1}}$ or
$\mathrm{NSS_{2}}$. The underlying reduced subscheme of $Z_{2}(M)$ can be
computed (see [BB22M2]) to be defined by the two by two minors of

$\begin{pmatrix}y_{0}&y_{1}&y_{2}\\\ y_{3}&y_{4}&y_{5}\end{pmatrix}.$

This is a Segre embedded $\mathbb{P}^{1}\times\mathbb{P}^{2}$ in
$\mathbb{P}^{5}$, i.e. three-dimensional and irreducible and not linear.
Therefore $M$ is also not of format $\mathrm{NSS_{3}}$ and thus $M$ is semi-
stable.

######  Theorem 4.9.

Let $M$ be of type $(a)$ to $(f)$ in Table 2. Then $M$ has the stability
property listed in the last column of Table 2.

###### Proof.

By Corollary 4.4 and Example 4.8 all matrices in Table 2 are semistable. If a
matrix with vanishing Pfaffian is strictly semistable, then by Proposition 3.3
it must be of type $(d)$, $(e)$ or $(f)$ in Table 2. Now by Proposition 4.2
the types $(a)-(f)$ are distinct. Consequently types $(a)$, $(b)$ and $(c)$
are stable.

Types $(d)$ and $(e)$ are not polystable, since type $(f)$ lies in the closure
of their orbits.

Finally matrices of type $(f)$ are polystable, since $d_{4}(M)$ is constant on
orbits and it can only decrease on the closure. Since $d_{4}(M)$ is minimal
for matrices of type $(f)$ this cannot happen. ∎

##  5\. appendix

Here we collect some known facts about formats.

matrix | format | action
---|---|---
| $2\times 2$
---
symmetric
$\left(\begin{smallmatrix}*&0\\\ 0&0\end{smallmatrix}\right)$ | $A\mapsto SAS^{t}$
$2\times 2$ | $\left(\begin{smallmatrix}*&0\\\ *&0\end{smallmatrix}\right)\quad\left(\begin{smallmatrix}*&*\\\ 0&0\end{smallmatrix}\right)$ | $A\mapsto SAT$
$3\times 3$ | $\left(\begin{smallmatrix}*&*&0\\\ *&*&0\\\ *&*&0\end{smallmatrix}\right)\quad\left(\begin{smallmatrix}*&*&*\\\ *&*&*\\\ 0&0&0\end{smallmatrix}\right)\quad\left(\begin{smallmatrix}*&*&*\\\ *&0&0\\\ *&0&0\end{smallmatrix}\right)\quad\left(\begin{smallmatrix}0&*&*\\\ *&0&*\\\ *&*&0\end{smallmatrix}\right)$ | $A\mapsto SAT$
| $4\times 4$
---
skew
$\left(\begin{smallmatrix}0&*&*&*\\\ *&0&0&0\\\ *&0&0&0\\\ *&0&0&0\end{smallmatrix}\right)\quad\left(\begin{smallmatrix}0&*&*&0\\\ *&0&*&0\\\ *&*&0&0\\\ 0&0&0&0\end{smallmatrix}\right).$ | $A\mapsto SAS^{t}$
Table 4. Formats of matrices of submaximal rank. The last format of the third
row and all formats of the last row are skew.

######  Theorem 5.1.

Let $M$ be a square matrix of linear forms with vanishing determinant. If $M$
is as in the first column of Table 4 then it has one of the formats depicted
in the second column of the same row.

###### Proof.

1st row: The determinant of a symmetric $2\times 2$ matrix defines a smooth
conic in the $\mathbb{P}^{2}$ of all symmetric $2\times 2$ matrices. The only
linear spaces contained in a smooth conic are points. These correspond to the
depicted format.

2nd row: The determinant of a $2\times 2$ matrix defines a smooth quadric
surface in the $\mathbb{P}^{3}$ of all $2\times 2$ matrices. The only linear
spaces contained in a smooth quadric surface are the lines of the two rulings.
These correspond to the two formats.

3rd row: This is a theorem by Testa [Te17].

4th row: The Pfaffian of a skew-symmetric $4\times 4$ matrix defines the
Grassmannian $G(2,4)$ in the space of all skew-symmetric $4\times 4$ matrices.
$G(2,4)$ parametrizes lines in a $\mathbb{P}^{3}$. The only linear spaces in
$G(2,4)$ are

  * •

For each point in $P\in\mathbb{P}^{3}$ the $\mathbb{P}^{2}$ of lines
containing this point

  * •

For each hyperplane $H\in\mathbb{P}^{3}$ the $\mathbb{P}^{2}$ of lines
contained in $H$

These two types correspond to the two formats depicted in the last row.

∎

## References

  * [BB22M2] Böhning, Chr., Graf v. Bothmer, H.C., _Macaulay 2 files for Formats of $6\times 6$ Skew Matrices of Linear Forms with Vanishing Pfaffian_, available at https://warwick.ac.uk/fac/sci/maths/people/staff/boehning/m2filesformats/
  * [Co21] Comaschi, G., _Stable linear systems of skew-symmetric forms of generic rank $\leq 4$_, Linear Algebra and its Applications 624 (2021),121–152
  * [FP98] Fulton, W., Pragacz, P., _Schubert varieties and degeneracy loci_ , Lecture Notes in Mathematics 1689, Springer (1998)
  * [MaMe05] Manivel, L., Mezzetti, E., _On linear spaces of skew-symmetric matrices of constant rank_ , Manuscr. Math. 117 (2005), 319–331
  * [Te17] Testa, D., _$3\times 3$ singular matrices of linear forms_, (2017), preprint, arXiv:1701.06864