# Identifiability and singular locus
of secant varieties to spinor varieties

Vincenzo Galgano111Dipartimento di Matematica, Università di Trento, Via
Sommarive 14 38123 Trento, Italy; ORCID: 0000-0001-8778-575X
<EMAIL_ADDRESS>
###### Abstract

In this work we analyze the $Spin$–structure of the secant variety of lines
$\sigma_{2}(\mathbb{S})$ to a Spinor variety $\mathbb{S}$ minimally embedded
in its spin representation. In particular, we determine the poset of the
$Spin$–orbits and their dimensions. We use it for solving the problems of
identifiability and tangential-identifiability in $\sigma_{2}(\mathbb{S})$,
and for determining the second Terracini locus of $\mathbb{S}$. Finally, we
show that the singular locus $Sing(\sigma_{2}(\mathbb{S}))$ contains the two
$Spin$–orbits of lowest dimensions and it lies in the tangential variety
$\tau(\mathbb{S})$: we also conjecture what it set-theoretically is.
Keywords: secant variety, Spinor variety, homogeneous space, Clifford
apolarity, identifiability, tangential-identifiability, Terracini locus,
singular locus.
Mathematics Subject Classification codes (2020): 14M17, 14N07, 15A66.

###### Contents

  1. 1 Preliminaries
  2. 2 Clifford apolarity
  3. 3 The poset of $\operatorname{Spin}_{2N}$–orbits in $\sigma_{2}(\mathbb{S}_{N}^{+})$
    1. 3.1 Secant orbits in $\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$
    2. 3.2 Tangent orbits in $\tau(\mathbb{S}_{N}^{+})$
    3. 3.3 Inclusions among closures of $\operatorname{Spin}_{2N}$–orbits
  4. 4 Identifiability in $\sigma_{2}(\mathbb{S}_{N}^{+})$
  5. 5 Tangential-identifiability in $\tau(\mathbb{S}_{N}^{+})$
  6. 6 Dimensions of $\operatorname{Spin}_{2N}$–orbits in $\sigma_{2}(\mathbb{S}_{N}^{+})$
  7. 7 The $2$–nd Terracini locus of $\mathbb{S}_{N}^{+}$
  8. 8 Results on the singular locus of $\sigma_{2}(\mathbb{S}_{N}^{+})$

## Acknowledgement

This work is part of the author’s Ph.D. thesis supervised by Alessandra
Bernardi and Giorgio Ottaviani, whom the author sincerely thanks for their
patient guidance and precious discussions. The author also thanks Laurent
Manivel and Mateusz Michałek for their helpful suggestions. The author is
member of the italian INdAM group GNSAGA.

## Introduction

Given an irreducible non-degenerate projective variety
$X\subset\mathbb{P}^{M}_{\mathbb{C}}$ and a point
$f\in\mathbb{P}^{M}_{\mathbb{C}}$, the $X$–rank of $f$ with respect to $X$ is
the minimum number of points of $X$ whose linear span contains $f$. In
particular, point of $X$ have $X$–rank $1$. Then, starting from $X$, for any
$r\in\mathbb{Z}_{>0}$ one defines the $r$–th secant variety
$\sigma_{r}(X)\subset\mathbb{P}^{M}_{\mathbb{C}}$ as the Zariski closure of
points of $\mathbb{P}^{M}_{\mathbb{C}}$ of $X$–rank at most $r$.
Secant varieties have been studied for decades, but several aspects of their
geometry are still mysterious and difficult to handle with. Even computing
their dimensions is a hard task and a current topic of research. Two crucial
aspects of secant varieties which are fundamental for applications and are
still unknown in general are the identifiability and the singularity of their
points. Namely, a given point $f\in\mathbb{P}^{M}_{\mathbb{C}}$ of $X$–rank
$r$ is identifiable if it admits a unique decomposition as sum of $r$ elements
of $X$. On the other hand, singular points in $\sigma_{r}(X)$ are those such
that the dimension of the tangent spaces at these points overcome the
dimension of the secant variety: conversely, the points for which such
dimensions coincide are smooth. Of course, identifiability and singularity are
of great impact to both theory and applications. Results having the smoothness
(ie. non-singularity) of the variety among the hypotheses are a dense in
geometry: for instance, in [COV14, Prop. 5.1] the authors give a criterion for
identifiability of specific tensors in a secant variety under the assumption
that these tensors are smooth. In fact, identifiability and smoothness are
quite related each other and often one notion suggests the other (as in this
work), although in general both implications admit counterexamples.
Owned to the above, determining the singular locus
$\operatorname{Sing}(\sigma_{r}(X))$ of a secant variety $\sigma_{r}(X)$ is a
central problem. Classically, if $\sigma_{r}(X)$ is not a linear space, it is
known that the singular locus $\operatorname{Sing}(\sigma_{r}(X))$ contains
the secant variety $\sigma_{r-1}(X)$ but only in few cases
$\operatorname{Sing}(\sigma_{r}(X))$ is actually determined. For instance, the
case $r=2$ for Segre varieties is solved in [MOZ15]; for Veronese varieties
the cases $r=2$ and $r=3$ are solved in [Kan99] and [Han18] respectively,
while partial results for higher cases $r\geq 4$ are obtained in [FH21];
partial results for the $2$–nd secant variety of Grassmannians and other
cominuscule varieties appear in [MM15].
Segre varieties, Veronese varieties and Grassmannians are example of
projective rational homogeneous varieties (RHV). They are described as
quotients $G/P$ of a semisimple (complex, in our case) Lie group $G$ by a
parabolic subgroup $P$, or equivalently as unique closed orbits (of the
highest weight vectors) into projectivized representations of such groups. The
Representation Theory behind these varieties allows to derive several
geometric properties: in this respect, the geometry of RHVs has been largely
studied in [LM03, LM04]. In particular, a wide literature has been devoted to
the secant variety of lines $\sigma_{2}(G/P)$ and the tangential variety
$\tau(G/P)$ (ie. unions of all tangent lines) to a RHV $G/P$, starting with
Zak’s key work [Zak93] and continuing with [Kaj99, LW07, LW09, MM15, Rus16].
Another example of RHVs is the Spinor varieties, described by the quotients
$\operatorname{Spin}(V)/P(v_{\omega})$ of a spin group
$\operatorname{Spin}(V)$ for the parabolic subgroup $P(v_{\omega})$
stabilizing the highest weight vector $v_{\omega}$ in a (half-)spin
representation of fundamental weight $\omega$. The (half-)spin representations
are defined by the fundamental weights corresponding to the right-hand-side
extremal nodes of the Dynkin diagrams of type $B_{N}$ and $D_{N}$.

$\dynkin[labels={\omega_{1},,,\omega_{N-1},\omega_{N}},edgelength=.75cm]B{oo.oo*}\
\ \ \ \ \ \ \ \ \
\dynkin[labels={\omega_{1},,,,\omega_{N-1},\omega_{N}},edgelength=.75cm]D{oo.oo**}$

Despite their bijection with maximal orthogonal Grassmannians, they are not
well understood as well as the classical Grassmannians. However, due to their
relation with Clifford algebras, they have a richer structure making them an
interesting topic of research. Moreover, Spinor varieties appear in Quantum
Information as sets of separable states in fermionic Fock spaces [LH15, LH18].
In general, equations and singular loci of secant varieties of lines to Spinor
varieties are still not determined: the only completely understood case is the
one of the Spinor variety defined by the spin group
$\operatorname{Spin}_{12}$, as one of the Legendrian varieties [LM01, LM07].
The aim of this paper is to describe the secant variety of lines to a Spinor
variety. Due to the fact that Spinor varieties of type $B_{N}$ and $D_{N+1}$
are projectively equivalent, throughout all this paper we work with the Spinor
variety $\mathbb{S}_{N}^{+}$ lying in the projectivization of the half-spin
representation $V_{\omega_{N}}^{D_{N}}=\bigwedge^{ev}\mathbb{C}^{N}$, under
the assumption that $N$ is even (cf. Remark 1.1). The main idea of the work is
to use the natural action of $\operatorname{Spin}_{2N}$ on
$V_{\omega_{N}}^{D_{N}}$, which leaves $\sigma_{2}(\mathbb{S}_{N}^{+})$
invariant. In particular, the singular locus
$\operatorname{Sing}(\sigma_{2}(\mathbb{S}_{N}^{+}))$ is stable under the
action of $\operatorname{Spin}_{2N}$, and it is the union of some orbits. Then
one can check the singularity or smoothness of an orbit by simply checking it
for a representative.
In Theorem 3.12 we prove that the poset of $\operatorname{Spin}_{2N}$–orbits
in $\sigma_{2}(\mathbb{S}_{N}^{+})$ is

$\mathbb{S}_{N}^{+}$$\Theta_{2,N}=\Sigma_{2,N}$$\Theta_{3,N}$$\vdots$$\Theta_{\frac{N}{2},N}$$\Sigma_{3,N}$$\vdots$$\Sigma_{\frac{N}{2},N}$

where the orbits $\Theta_{l,N}$ (cf. (3.6)) are made of tangent points to
$\mathbb{S}_{N}^{+}$, the orbits $\Sigma_{l,N}$ (cf. (3.1)) are made of points
lying on bisecant lines to $\mathbb{S}_{N}^{+}$, and the arrows denote the
inclusion of an orbit into the closure of another orbit.
Via the theory of the nonabelian apolarity from [LO13], we introduce the
Clifford apolarity (cf. Sec. 2) which allow us to completely solve the
problems of identifiability and tangential-identifiability (cf. Def. 1.5) of
points in $\sigma_{2}(\mathbb{S}_{N}^{+})$. More precisely, we prove the
following result, collecting Proposition 4.3 and Theorems 4.5, 5.2 at once.

######  Main Theorem.

Points in the orbit $\Sigma_{2,N}$ are not identifiable, while points in the
orbits $\Sigma_{l,N}$ for $l\geq 3$ are identifiable. Moreover, points in the
orbits $\Theta_{l,N}$ for $l\geq 3$ are tangential-identifiable.

We apply these results for computing the dimensions of the orbit closures.
Moreover, we describe the second Terracini locus of a Spinor variety, whose
importance in geenral relies in its strong relation with the singular locus.

######  Theorem (Theorem 7.1).

The second Terracini locus $\operatorname{Terr}_{2}(\mathbb{S}_{N}^{+})$ of
the Spinor variety $\mathbb{S}_{N}^{+}$ corresponds to the orbit closure
$\overline{\Sigma_{2,N}}=\mathbb{S}_{N}^{+}\sqcup\Sigma_{2,N}$.

Finally, we obtain a lower and upper bound on the singular locus of
$\sigma_{2}(\mathbb{S}_{N}^{+})$.

######  Theorem (Corollary 8.3).

For any $N\geq 7$, the singular locus of the secant variety of lines
$\sigma_{2}(\mathbb{S}_{N}^{+})$ contains the orbit closure
$\overline{\Sigma_{2,N}}$ and lies in the tangential variety
$\tau(\mathbb{S}_{N}^{+})$.

We also conjecture that the singular locus actually coincides with the above
orbit closure.
We point out that the same poset of orbits and the same results on
identifiability have been obtained in [GS23] for the case of Grassmannians,
simultaneously to this work, although with different techniques. This suggests
a more general behaviour for a wider subclass of RHVs, namely the cominuscule
varieties. However, we leave this generalization for future work.
The paper is organized as follows. In Sec. 1 we recall some basic facts and we
fix our notation. Sec. 2 is devoted to the Clifford apolarity. In Sec. 3 we
determine the poset of $\operatorname{Spin}_{2N}$–orbits in
$\sigma_{2}(\mathbb{S}_{N}^{+})$. In Secc. 4 and 5 we apply the Cliffor
apolarity for determining identifiable and tangential-identifiable points in
$\sigma_{2}(\mathbb{S}_{N}^{+})$: these results allow us to compute the
dimensions of the orbit closures in Sec. 6, as well as to describe the second
Terracini locus $\operatorname{Terr}_{2}(\mathbb{S}_{N}^{+})$ in Sec. 7.
Finally, in Sec. 8 we give bounds on the singular locus
$\operatorname{Sing}(\sigma_{2}(\mathbb{S}_{N}^{+}))$ and we conjecture what
it actually we expect it to be. The appendix contains arguments suggesting
that our conjecture should be true.

## Notation

For any integers $a\leq b\leq k\in\mathbb{Z}_{>0}$ we fix the notation

$[k]:=\\{1,\ldots,k\\}\ \ \ \ ,\ \ \ \ \binom{[k]}{a}:=\left\\{I\subset[k]\ |\
|I|=a\right\\}\ ,$

and $i=a:b$ meaning $i\in\\{a,a+1,\ldots,b\\}$. Given $(e_{1},\ldots,e_{N})$
the standard basis of $\mathbb{C}^{N}$, we denote by
$\mathbb{e}_{I}:=e_{i_{1}}\wedge\ldots\wedge
e_{i_{k}}\in\bigwedge^{k}\mathbb{C}^{N}$ the exterior product of the basis
vectors indexed by the subset $I\in\binom{[N]}{k}$.

##  1 Preliminaries

We refer to [Pro07, Ch. 5, Secc. 4-5] for details on Clifford algebras and
spin groups, and [Pro07, Ch. 11, Sec. 7] for spin representations. Since
Spinor varieties of Dynkin type $B_{N}$ and $D_{N+1}$ are projectively
equivalent, they share the same geometry inside their minimal homogeneous
embedddings. In this respect, we only work with spin groups of Dynkin type $D$
defined starting from even-dimensional vector spaces.

#### Clifford algebras and spin groups.

Given a finite-dimensional vector space $V$ over $\mathbb{C}$ endowed with a
non-degenerate quadratic form $q\in\operatorname{Sym}^{2}V^{\vee}$, we denote
by $\operatorname{Cl}_{q}(V)$ the Clifford algebra of $q$ over $V$ and by
$\operatorname{Cl}_{q}^{+}(V)$ its even-degree component. One can always
reduce to study Clifford algebras over even-dimensional vector spaces, thus we
consider $\dim V=2N$.
We fix the hyperbolic standard basis $(e_{1},\ldots,e_{N},f_{1},\ldots,f_{N})$
of $V$ with respect to the quadratic form $q=\sum_{k=1}^{N}x_{k}x_{N+k}$, so
that the orthogonal decomposition $V=E\oplus E^{\vee}$ holds for $E=\langle
e_{1},\ldots,e_{N}\rangle_{\mathbb{C}}$ and $E^{\vee}=\langle
f_{1},\ldots,f_{N}\rangle_{\mathbb{C}}$ fully isotropic subspaces of maximal
dimension $\lfloor\frac{2N}{2}\rfloor=N$. Since
$\bigwedge^{\bullet}V\simeq\operatorname{Cl}_{q}(V)$ (as vector spaces) [LM89,
Sec. 1.1, Proposition 1.3], the Clifford multiplication in
$\operatorname{Cl}_{q}(V)$ can be described as follows: for any $v=e+f\in
E\oplus E^{\vee}$ and $x\in\operatorname{Cl}_{q}(V)\simeq\bigwedge^{\bullet}V$
it holds

$v\cdot x\simeq e\wedge x+f\neg x\ ,$ (1.1)

where, for $x=x_{1}\wedge\ldots\wedge x_{k}\in\bigwedge^{k}V$, $f\neg
x:=\sum_{i=1}^{k}(-1)^{i+1}q(f,x_{i})x_{1}\wedge\ldots\wedge\widehat{x_{i}}\wedge\ldots\wedge
x_{k}\in\bigwedge^{k-1}V$.
For any invertible element $x=v_{1}\cdots
v_{k}\in\operatorname{Cl}_{q}(V)^{\times}$, its spinor norm and its reverse
involution are respectively

$N(x):=xx^{*}=q(v_{1})\cdots q(v_{k})\in\mathbb{C}^{\times}\ \ \ ,\ \ \
x^{*}:=v_{k}\cdots v_{1}\ .$ (1.2)

Then the spin group is
$\operatorname{Spin}(V,q):=\left\\{x\in\operatorname{Cl}_{q}^{+}(V)^{\times}\
|\ xVx^{-1}=V,\ N(x)=1\right\\}$. For any $x\in\operatorname{Spin}(V,q)$ it
holds $\big{(}\operatorname{conj}_{x}:v\mapsto
xvx^{-1}\big{)}\in\operatorname{SO}(V,q)$. We write $\operatorname{Spin}_{2N}$
for the spin group $\operatorname{Spin}(\mathbb{C}^{2N},q)$ of Dynkin type
$D_{N}$.

#### Spin representations.

In the same notation as above, for $V=E\oplus E^{\vee}$ of dimension $\dim
V=2N$, the half-spin representations $V_{\omega_{N-1}}^{D_{N}}$ and
$V_{\omega_{N}}^{D_{N}}$ for $\operatorname{Spin}_{2N}$ are defined by the two
fundamental weights $\omega_{N-1}$ and $\omega_{N}$ on the right-hand-side
extremal nodes of the Dynkin diagram of type $D_{N}$:

$\dynkin[labels={,,,,\omega_{N-1},\omega_{N}},edgelength=.75cm]D{oo.oo**}$

They coincide with the $2^{N-1}$–dimensional vector spaces $\bigwedge^{{ev}}E$
and $\bigwedge^{{od}}E$ depending on the parity of $N$. More precisely, given
$v_{\omega_{i}}$ and $\ell_{\omega_{i}}$ the highest and lowest weight vectors
of the irreducible representation $V_{\omega_{i}}^{D_{N}}$ respectively, it
holds:

$N\equiv 0$ (mod $2$)
---
|  $V_{\omega_{i}}^{D_{N}}$ |  $v_{\omega_{i}}$ |  $\ell_{\omega_{i}}$
$\omega_{N-1}$ |  $\bigwedge^{od}E$ |  $\mathbb{e}_{[N-1]}=e_{1}\wedge\ldots\wedge e_{N-1}$ |  $e_{1}$
$\omega_{N}$ |  $\bigwedge^{ev}E$ |  $\mathbb{e}_{[N]}=e_{1}\wedge\ldots\wedge e_{N}$ |  $\mathbbm{1}$
Table 1: Half-spin representations of $\operatorname{Spin}_{2N}$ for $N\equiv
0$ (mod $2$). $N\equiv 1$ (mod $2$)
---
|  $V_{\omega_{i}}^{D_{N}}$ |  $v_{\omega_{i}}$ |  $\ell_{\omega_{i}}$
$\omega_{N-1}$ |  $\bigwedge^{ev}E$ |  $\mathbb{e}_{[N-1]}=e_{1}\wedge\ldots\wedge e_{N-1}$ |  $\mathbbm{1}$
$\omega_{N}$ |  $\bigwedge^{od}E$ |  $\mathbb{e}_{[N]}=e_{1}\wedge\ldots\wedge e_{N}$ |  $e_{1}$
Table 2: Half-spin representations of $\operatorname{Spin}_{2N}$ for $N\equiv
1$ (mod $2$).

where we write the scalar $\mathbbm{1}:=1\in\mathbb{C}$ when considered as
weight vector (or simply as element of the representation). The half-spin
representations $V_{\omega_{N-1}}^{D_{N}}$ and $V_{\omega_{N}}^{D_{N}}$ are
self-dual if $N$ is even, and dual one to the other if $N$ is odd.

#### Spinor varieties.

The Spinor varieties $\mathbb{S}_{N}^{+}\subset\mathbb{P}(\bigwedge^{ev}E)$
and $\mathbb{S}_{N}^{-}\subset\mathbb{P}(\bigwedge^{od}E)$ are the rational
homogeneous varieties
$\operatorname{Spin}_{2N}\cdot[v_{\omega_{i}}]=\operatorname{Spin}_{2N}/P_{\omega_{i}}\subset\mathbb{P}(V_{\omega_{i}}^{D_{N}})$
for $P_{\omega_{i}}$ being the parabolic subgroup corresponding to the
fundamental weight $\omega_{i}$, or equivalently the stabilizer of
$v_{\omega_{i}}$. They have dimension $\binom{N}{2}$ and they are locally
parametrized by the set of skew-symmetric matrices
$\bigwedge^{2}\mathbb{C}^{N}$. Points in $V_{\omega_{i}}^{D_{N}}$ are said
spinors, while the ones in $\mathbb{S}_{N}^{\pm}$ are said pure spinors.

######  Remark.

For $V=\mathbb{C}u\oplus E\oplus E^{\vee}$ of odd dimension $2N+1$, the
$B_{N}$–type spin group $\operatorname{Spin}_{2N+1}$ has only one spin-
representation $V_{\omega_{N}}^{B_{N}}=\bigwedge^{\bullet}Eu$, hence only one
Spinor variety $\mathbb{S}_{N}\subset\mathbb{P}(\bigwedge^{\bullet}Eu)$.

######  Remark 1.1 (Setting).

The Spinor variety for $\operatorname{Spin}(2N+1)$ and the Spinor varieties
for $\operatorname{Spin}(2N+2)$ are projectively equivalent. Then from now on
we work only with $V=E\oplus E^{\vee}$ of dimension $\dim V=2N$ for an even
$N$, hence with the Spinor variety
$\mathbb{S}_{N}^{+}=\operatorname{Spin}_{2N}\cdot[\mathbb{e}_{[N]}]\subset\mathbb{P}\left(\bigwedge^{ev}E\right)$.
All the arguments of this work can be repeated for $\dim V\equiv 2$ (mod $4$)
with the Spinor variety
$\mathbb{S}_{N}^{-}=\operatorname{Spin}_{2N}\cdot[\mathbb{e}_{[N]}]\subset\mathbb{P}(\bigwedge^{od}E)$.

#### Maximal orthogonal Grassmannians.

For any $a\in\bigwedge^{\bullet}E$, the Clifford multiplication (1.1) defines
a map

$\begin{matrix}\psi_{a}:&V&\longrightarrow&\bigwedge^{\bullet}E\\\
&v&\mapsto&v\cdot a\end{matrix}$ (1.3)

whose kernel $H_{a}:=\operatorname{ker}(\psi_{a})$ is always fully isotropic
(i.e. $H_{a}\subset H_{a}^{\perp}$). Moreover, $H_{a}$ is of maximum dimension
$\dim H_{a}=N$ (i.e. $H_{a}=H_{a}^{\perp}$) if and only if
$[a]\in\mathbb{S}_{N}^{+}$. Notice that, in the notation from the previous
paragraph, one has $H_{\mathbb{e}_{[N]}}=E$ and $H_{\mathbbm{1}}=E^{\vee}$.
The set $\operatorname{OGr}(N,V)$ of maximal fully isotropic subspaces of $V$
has two connected components, called maximal orthogonal Grassmannians:

$\displaystyle\operatorname{OGr}^{+}(N,V)$
$\displaystyle:=\left\\{H\in\operatorname{OGr}(N,V)\ |\
\operatorname{codim}_{E}(H\cap E)\equiv 0\ (\text{mod}\ 2)\right\\}\ ,$
$\displaystyle\operatorname{OGr}^{-}(N,V)$
$\displaystyle:=\left\\{H\in\operatorname{OGr}(N,V)\ |\
\operatorname{codim}_{E}(H\cap E)\equiv 1\ (\text{mod}\ 2)\right\\}\ .$

Every $H\in\operatorname{OGr}(N,V)$ defines a $1$–dimensional subspace
$\left\\{\varphi\in\bigwedge^{\bullet}E\ |\ w\cdot\varphi=0\ \forall w\in
H\right\\}=\langle a_{H}\rangle_{\mathbb{C}}\subset\bigwedge^{\bullet}E$, and
$a_{H}$ is a pure spinor in either one of the two half-spin representations,
depending on the parity of $N$. In particular, one has the one-to-one
correspondence

$\begin{matrix}\mathbb{S}_{N}^{+}\ \cup\
\mathbb{S}_{N}^{-}&\stackrel{{\scriptstyle
1:1}}{{\longleftrightarrow}}&\operatorname{OGr}^{+}(N,V)\ \cup\
\operatorname{OGr}^{-}(N,V)\\\
[a]&\longrightarrow&H_{a}=\operatorname{ker}(\psi_{a})\\\
[a_{H}]&\longleftarrow&H\end{matrix}\ .$ (1.4)

Notice that every non pure spinor
$a\in(\bigwedge^{\bullet}E)\setminus(\mathbb{S}_{N}^{+}\sqcup\mathbb{S}_{N}^{-})$
defines a non maximal fully isotropic subspace
$H_{a}=\operatorname{ker}(\psi_{a})$, but the latter does not define a unique
spinor (cf. [Bat14, Sec. 3, Theorem 1]). We exhibit the association
$H\mapsto[a_{H}]$. Given $H\in\operatorname{OGr}^{+}(N,V)$ with $\dim(H\cap
E)=p$ even, up to action of $\operatorname{SO}(V)$ one can assume $H=\langle
e_{1},\ldots,e_{p},g_{p+1},\ldots,g_{N}\rangle_{\mathbb{C}}$ where

$g_{j}=f_{j}+\sum_{k=p+1}^{N}\alpha_{kj}e_{k}\ \ \ ,\forall j=p+1:N$ (1.5)

for a suitable skew-symmetric matrix
$(\alpha_{kj})\in\bigwedge^{2}\mathbb{C}^{N-p}$. Then via (1.4) $H$
corresponds to a pure spinor $[a_{H}]\in\mathbb{S}_{N}^{+}$ such that $h\cdot
a_{H}=0$ for any $h\in H$. The following result follows by a direct
computation of the Clifford actions $g_{j}\cdot a_{H}=0$ for any $j=p+1:N$.

######  Proposition 1.2.

Let $N,p$ be even. Let $H=\langle
e_{1},\ldots,e_{p},g_{p+1},\ldots,g_{N}\rangle_{\mathbb{C}}\in\operatorname{OGr}^{+}(N,V)$,
where the $g_{j}$’s are as in (1.5) and define the skew-symmetric matrix
$A=(\alpha_{kj})_{k,j}\in\bigwedge^{2}\mathbb{C}^{N-p}$. Then $H$ corresponds
to the pure spinor

$a_{H}=\sum_{I\subset\\{p+1,\ldots,N\\}}\operatorname{Pf}(A_{I})\mathbb{e}_{[p]}\wedge\mathbb{e}_{I}\
,$

where $\operatorname{Pf}(A_{I})$ is the pfaffian of the submatrix $A_{I}$ of
$A$ whose rows and columns are indexed by $I$, and we set
$\mathbb{e}_{\emptyset}:=\mathbbm{1}$.

#### Hamming distance of varieties.

Given $X\subset\mathbb{P}^{M}$ a projective variety, the Hamming distance
$d(p,q)$ between two points $p,q\in X$, already considered in [BDD07, AOP12,
GS23], is the minimum number of lines of $\mathbb{P}^{M}$ lying in $X$ and
connecting $p$ to $q$, that is

$d(p,q):=\min\left\\{r\in\mathbb{N}\ |\ \exists\text{ distinct
}p_{1},\ldots,p_{r-1}\in X,\ L(p_{i},p_{i+1})\subset X\ \forall
i=0:r-1\right\\}$ (1.6)

where we set $p_{0}=p$, $p_{r}=q$ and $L(p_{i},p_{i+1})$ is the line passing
through $p_{i}$ and $p_{i+1}$. In particular, $d(p,q)=0$ if and only if $p=q$,
while $d(p,q)=1$ if and only if $L(p,q)\subset X$ and $p\neq q$. The diameter
of $X$ is $\operatorname{diam}(X):=\max_{p,q\in X}\left\\{d(p,q)\right\\}$.

######  Remark 1.3.

Given $G$ a group, $W$ a representation of $G$ and $X\subset\mathbb{P}(W)$ a
variety which is invariant under the $G$-action induced on $\mathbb{P}(W)$,
then the $G$-action preserves the Hamming distance between points in $X$, that
is $d(g\cdot p,g\cdot q)=d(p,q)$.

For certain rational homogeneous varieties, called cominuscule, (among which,
the Spinor varieties of $D$–type) the above distance coincides with the
minimum possible degree of rational curves passing through the two points
[Buc+13, Lemma 4.2]. From the proof of [Buc+13, Proposition 4.5] we get the
following result.

######  Proposition 1.4.

Let $[a],[b]\in\mathbb{S}_{N}^{+}$ be two pure spinors and let
$H_{a},H_{b}\in\operatorname{OGr}^{+}(N,V)$ be their corresponding maximal
fully isotropic subspaces. Then

$d([a],[b])=\frac{\operatorname{codim}_{H_{a}}(H_{a}\cap H_{b})}{2}\ .$

As a consequence, for any $N\geq 2$ the Spinor varieties $\mathbb{S}_{N}^{+}$
have diameter $\left\lfloor\frac{N}{2}\right\rfloor$ (cf. table in [Buc+13,
Sec. 4]). Finally, as Spinor varieties are intersection of quadrics, for any
two distinct points $[a],[b]\in\mathbb{S}_{N}^{+}$ by Bèzout one gets
$d([a],[b])\geq 2$ if and only if
$L([a],[b])\cap\mathbb{S}_{N}^{+}=\\{[a],[b]\\}$.

#### Secant varieties.

The secant variety of lines
$\sigma_{2}(\mathbb{S}_{N}^{+})\subset\mathbb{P}(\bigwedge^{ev}E)$ of the
Spinor variety $\mathbb{S}_{N}^{+}\subset\mathbb{P}(\bigwedge^{ev}E)$ is the
union of all secant and tangent lines to $\mathbb{S}_{N}^{+}$ in
$\mathbb{P}(\bigwedge^{ev}E)\simeq\mathbb{P}^{2^{N-1}-1}$. We denote by
$\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+}):=\\{[a+b]\in\sigma_{2}(\mathbb{S}_{N}^{+})\
|\ [a],[b]\in\mathbb{S}_{N}^{+}\\}$ the set of all points in
$\sigma_{2}(\mathbb{S}_{N}^{+})$ lying on a bisecant line to
$\mathbb{S}_{N}^{+}$. As a rational homogeneous variety, it is known that
$\sigma_{2}(\mathbb{S}_{N}^{+})=\overline{\operatorname{Spin}_{2N}\cdot[v_{\omega_{i}}+\ell_{\omega_{i}}]}$
is quasi-homogeneous [Zak93, Theorem 1.4] and non-defective for any $N$
[Kaj99, Ang11], that is

$\dim\sigma_{2}(\mathbb{S}_{N}^{+})=\min\left\\{2\dim\mathbb{S}_{N}^{+}+1,\dim\mathbb{P}\left(\bigwedge^{ev}E\right)\right\\}=\begin{cases}2^{N-1}-1&\text{for
}N\leq 6\\\ N(N-1)+1&\text{for }N\geq 6\end{cases}\ .$ (1.7)

For $N\leq 5$ the secant variety $\sigma_{2}(\mathbb{S}_{N}^{+})$ overfills
the ambient space $\mathbb{P}^{2^{N-1}-1}$ (i.e.
$\dim\sigma_{2}(\mathbb{S}_{N}^{+})=2^{N-1}-1\lneq
2\dim\mathbb{S}_{N}^{+}+1$), for $N\geq 7$ it is strictly contained, while for
$N=6$ it perfectly fills $\mathbb{P}^{31}$ (in the sense
$\dim\sigma_{2}(\mathbb{S}_{N}^{+})=2\dim(\mathbb{S}_{N}^{+})+1=2^{N-1}-1$):
the latter case is said to be perfect.

#### Identifiability.

For any $[p]\in\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$, the decomposition
locus of $p$ is the set of all pairs of pure spinors in $\mathbb{S}_{N}^{+}$
giving a decomposition of $[p]$, that is

$Dec_{\mathbb{S}_{N}^{+}}([p]):=\left\\{([a],[b])\ |\
[a],[b]\in\mathbb{S}_{N}^{+},\
[p]\in\langle[a],[b]\rangle\right\\}\subset(\mathbb{S}_{N}^{+})^{2}_{/\mathfrak{S}_{2}}\
,$

where $(\mathbb{S}_{N}^{+})^{2}_{/\mathfrak{S}_{2}}$ denotes the symmetric
quotient of $(\mathbb{S}_{N}^{+})^{2}$ by the symmetric group
$\mathfrak{S}_{2}$. An element $([a],[b])\in Dec_{\mathbb{S}_{N}^{+}}([p])$ is
called a decomposition of $[p]$.
A point $[p]\in\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$ is identifiable if
$Dec_{\mathbb{S}_{N}^{+}}([p])$ is a singleton. Geometrically, this means that
$[p]$ lies on a unique bisecant line to $\mathbb{S}_{N}^{+}$. Otherwise one
says that $[p]$ is unidentifiable. For any subset
$Y\subset\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$, we say that $Y$ is
(un)identifiable if any point of $Y$ is so. In particular, an orbit is
(un)identifiable if and only if a representative of it is so.

#### Tangential-identifiability

The tangential variety of $\mathbb{S}_{N}^{+}$ in
$\mathbb{P}(\bigwedge^{ev}E)$ is the union of all lines in
$\mathbb{P}(\bigwedge^{ev}E)$ which are tangent to $\mathbb{S}_{N}^{+}$, i.e.
$\tau(\mathbb{S}_{N}^{+}):=\bigcup_{p\in\mathbb{S}_{N}^{+}}T_{p}\mathbb{S}_{N}^{+}\
\subset\mathbb{P}(\bigwedge^{ev}E)$. In particular, it holds
$\mathbb{S}_{N}^{+}\subset\tau(\mathbb{S}_{N}^{+})\subset\sigma_{2}(\mathbb{S}_{N}^{+})$,
and
$\sigma_{2}(\mathbb{S}_{N}^{+})=\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})\cup\tau(\mathbb{S}_{N}^{+})$
(the union may be non-disjoint). Moreover, from Zak’s key result [Zak93,
Theorem 1.4] it holds that either
$\dim\tau(\mathbb{S}_{N}^{+})=2\dim\mathbb{S}_{N}^{+}$ and
$\dim\sigma_{2}(\mathbb{S}_{N}^{+})=2\dim\mathbb{S}_{N}^{+}+1$, or
$\tau(\mathbb{S}_{N}^{+})=\sigma_{2}(\mathbb{S}_{N}^{+})$.

######  Definition 1.5.

A tangent point $[q]\in\tau(\mathbb{S}_{N}^{+})$ is tangential-identifiable if
it lies on a unique tangent line to $\mathbb{S}_{N}^{+}$, or equivalently if
there exists a unique $[p]\in\mathbb{S}_{N}^{+}$ such that $[q]\in
T_{[p]}\mathbb{S}_{N}^{+}$. Otherwise it is tangential-unidentifiable.

We say that an orbit is tangential-(un)identifiable if all of its elements are
so.

######  Definition 1.6.

Given a tangent point $[q]\in\tau(\mathbb{S}_{N}^{+})$, its tangential-locus
is the set of points $p\in\mathbb{S}_{N}^{+}$ such that $q\in
T_{p}\mathbb{S}_{N}^{+}$.

Clearly, if $q$ is tangential-identifiable, then its tangential-locus is given
by a single point at $\mathbb{S}_{N}^{+}$.

#### Nonabelian apolarity.

The classical apolarity action [IK99] and the skew-apolarity action [Arr+21,
Definition 4] are particular cases of a more general apolarity, namely the
nonabelian apolarity, introduced at first in [LO13, Section 1.3]. Let
$X\subset\mathbb{P}^{n}$ be a projective variety and let $\mathcal{L}$ be a
very ample line bundle on $X$ giving the embedding
$X\subset\mathbb{P}^{n}=\mathbb{P}(H^{0}(X,\mathcal{L})^{\vee})$. Let
$\mathcal{E}$ be a rank–$e$ vector bundle on $X$ such that
$H^{0}(X,\mathcal{E}^{\vee}\otimes\mathcal{L})$ is not trivial, where
$\mathcal{E}^{\vee}$ denotes the bundle dual to $\mathcal{E}$. The natural
contraction map $H^{0}(X,\mathcal{E})\otimes
H^{0}(X,\mathcal{E}^{\vee}\otimes\mathcal{L})\longrightarrow
H^{0}(X,\mathcal{L})$ leads to the morphism $A:H^{0}(X,\mathcal{E})\otimes
H^{0}(X,\mathcal{L})^{\vee}\longrightarrow
H^{0}(X,\mathcal{E}^{\vee}\otimes\mathcal{L})^{\vee}$ and, after fixing a
given $f\in H^{0}(X,\mathcal{L})^{\vee}$, one gets the linear map

$A_{f}:H^{0}(X,\mathcal{E})\longrightarrow
H^{0}(X,\mathcal{E}^{\vee}\otimes\mathcal{L})^{\vee}$

defined as $A_{f}(s):=A(s\otimes f)$ for any $s\in H^{0}(X,\mathcal{E})$, and
called nonabelian apolarity action. Let
$H^{0}(X,\mathcal{I}_{Z}\otimes\mathcal{E})$ and
$H^{0}(X,\mathcal{I}_{Z}\otimes\mathcal{E}^{\vee}\otimes\mathcal{L})$ be the
spaces of global sections vanishing on a $0$–dimensional subscheme $Z\subset
X$, and let $\langle Z\rangle\subset H^{0}(X,\mathcal{L})^{\vee}$ be the
linear span of $Z$. For a proof of the following result we refer to [LO13,
Proposition 5.4.1].

######  Proposition 1.7.

Let $f\in H^{0}(X,\mathcal{L})^{\vee}$ and let $Z\subset X$ be a
$0$–dimensional subscheme such that $f\in\langle Z\rangle$. Then it holds
$H^{0}(X,\mathcal{I}_{Z}\otimes\mathcal{E})\subseteq\operatorname{ker}A_{f}$
and
$H^{0}(X,\mathcal{I}_{Z}\otimes\mathcal{E}^{\vee}\otimes\mathcal{L})\subseteq\operatorname{Im}A_{f}^{\perp}$.

It is worth remarking that the above result holds for non-reduced subschemes
$Z\subset X$ too. The schematic non-reduced version of the nonabelian
apolarity has already been considered in [OR20, Theorem 6.10] for Veronese
varieties, and more generally in [Gał17, Proposition 7]. In the classical
apolarity, the case of minimal subschemes is considered with respect to cactus
varieties and cactus rank [BB14, BR13], while for non-minimal subschemes see
[BT20]. Finally, for reduced subschemes we recall the following result [OO13,
Proposition 4.3].

######  Proposition 1.8.

Let $f\in H^{0}(X,\mathcal{L})^{\vee}$ and let $Z\subset X$ be a
$0$–dimensional reduced subscheme of length $r$, minimal with respect to the
property $f\in\langle Z\rangle$. Assume that
$\operatorname{Rk}A_{f}=r\cdot\operatorname{Rk}\mathcal{E}$. Then it holds
$H^{0}(X,\mathcal{I}_{Z}\otimes\mathcal{E})=\operatorname{ker}A_{f}$ and
$H^{0}(X,\mathcal{I}_{Z}\otimes\mathcal{E}^{\vee}\otimes\mathcal{L})=\operatorname{Im}A_{f}^{\perp}$.
In particular, $Z$ is contained in the common zero locus of
$\operatorname{ker}A_{f}$ and $\operatorname{Im}A_{f}^{\perp}$.

##  2 Clifford apolarity

In this section we analyze the nonabelian apolarity in the case of Spinor
varieties, obtaining what we call Clifford apolarity. We also exhibit the
vanishing condition of the global sections of a certain vector bundle on
$\mathbb{S}_{N}^{+}$. We assume the setting in Remark 1.1.
Consider the half-spin representations
$V_{\omega_{N-1}^{D_{N}}}=\bigwedge^{od}E$ and
$V_{\omega_{N}}^{D_{N}}=\bigwedge^{ev}E$, and consider the standard
representation $V_{\omega_{1}}^{D_{N}}=V$. Let $\mathcal{O}(1)$ be the very
ample line bundle $\mathcal{O}(1)$ on $\mathbb{S}_{N}^{+}$. Via the
identification with $\operatorname{OGr}^{+}(N,V)$, let $\mathcal{U}$ be the
rank–$N$ universal bundle on $\mathbb{S}_{N}^{+}$ obtained as the pullback of
the universal bundle on $\operatorname{Gr}(N,V)$. The bundles
$\mathcal{O}(1)$, $\mathcal{U}^{\vee}$ and
$\mathcal{U}(1):=\mathcal{U}\otimes\mathcal{O}(1)$ on $\mathbb{S}_{N}^{+}$ are
homogeneous and the corresponding representations of the parabolic subgroup
$P_{\omega_{N}}$ are irreducible with heighest weights $\omega_{N}$,
$\omega_{1}$ and $\omega_{N-1}$ respectively: from Borel–Weil’s Theorem one
gets

$H^{0}\big{(}\mathbb{S}_{N}^{+},\mathcal{O}(1)\big{)}\simeq\left(\bigwedge^{ev}E\right)^{\vee}\
\ \ ,\ \ \
H^{0}\big{(}\mathbb{S}_{N}^{+},\mathcal{U}(1)\big{)}\simeq\left(\bigwedge^{od}E\right)^{\vee}\
\ \ \,\ \ \ H^{0}\big{(}\mathbb{S}_{N}^{+},\mathcal{U}^{\vee}\big{)}\simeq
V^{\vee}\ .$

The contraction map $H^{0}(\mathbb{S}_{N}^{+},\mathcal{U}^{\vee})\otimes
H^{0}(\mathbb{S}_{N}^{+},\mathcal{U}(1))\rightarrow
H^{0}(\mathbb{S}_{N}^{+},\mathcal{O}(1))$ is equivalent to

$\left(V_{\omega_{1}}^{D_{N}}\right)^{\vee}\otimes\left(V_{\omega_{N-1}}^{D_{N}}\right)^{\vee}\longrightarrow\left(V_{\omega_{N}}^{D_{N}}\right)^{\vee}\
,$

and, from the duality of half-spin representations (as $N$ is even), we get

$\left(E\oplus
E^{\vee}\right)\otimes\bigwedge^{od}E\longrightarrow\bigwedge^{ev}E\ .$ (2.1)

Such map is uniquely determined as $\operatorname{Spin}(V)$-equivariant
morphism (up to scalars): this is a consequence of Schur’s lemma applied to
the following result.

######  Theorem 2.1.

The irreducible $\operatorname{Spin}(V)$-module $\bigwedge^{ev}E$ (resp.
$\bigwedge^{od}E$) appears with multiplicity $1$ in the
$\operatorname{Spin}(V)$-module $V\otimes\bigwedge^{od}E$ (resp.
$V\otimes\bigwedge^{ev}E$).

###### Proof.

In light of the natural inclusion
$\operatorname{SL}(E)\subset\operatorname{Spin}(V)$, for any
$\operatorname{Spin}(V)$-module $W$ among the above ones we can consider its
restriction as $\operatorname{SL}(E)$-module
$\operatorname{Res}_{\operatorname{SL}(E)}^{\operatorname{Spin}(V)}(W)$: we
lighten up the notation by simply writing $W$ and specifying its module
structure. Since $N$ is even, one can rewrite the
$\operatorname{SL}(E)$–module $(E\oplus E^{\vee})\otimes\bigwedge^{od}E$ as

$(E\oplus
E^{\vee})\otimes\bigwedge^{od}E=\bigoplus_{k=0}^{\frac{N-2}{2}}\left(E\otimes\bigwedge^{2k+1}E\right)\oplus\left(E^{\vee}\otimes\bigwedge^{2k+1}E\right)\
.$

From Pieri’s formula [Pro07, Sec. 9.10.2], $(E\oplus
E^{\vee})\otimes\bigwedge^{od}E$ decomposes into the
$\operatorname{SL}(E)$-modules

$V\otimes\bigwedge^{od}E=\mathbb{C}\oplus
2\left(\bigoplus_{k=1}^{\frac{N-4}{2}}\bigwedge^{2k}E\right)\oplus\bigwedge^{N}E\oplus\left(\bigoplus_{k=0}^{\frac{N-2}{2}}S^{\scriptsize\overbrace{(2,1,\ldots,1)}^{2k+1}}\oplus
S^{\scriptsize(\overbrace{2,\ldots,2}^{2k+1},\overbrace{1,\ldots,1}^{N-2-2k})}\right)\
,$

among which there is only one copy of $\bigwedge^{ev}E$. Thus there has to be
only one copy in the $\operatorname{Spin}(V)$-module decomposition as well. ∎

By fixing $q\in\bigwedge^{ev}E$, from the map (2.1) we get a
$\operatorname{Spin}(V)$-equivariant map $\psi_{q}:(E\oplus
E^{\vee})\rightarrow\bigwedge^{od}E$ which is unique (up to scalars) by
Theorem 2.1, thus it coincides with the map $\psi_{q}:V\rightarrow\bigwedge E$
defined in (1.3). Moreover, the natural contraction (2.1) is equivalent to a
map

$\Phi:\bigwedge^{ev}E\otimes\bigwedge^{od}E^{\vee}\longrightarrow E\oplus
E^{\vee}$

which again by Theorem 2.1 is uniquely determined as
$\operatorname{Spin}(V)$-equivariant morphism: more precisely, it is the
projection onto the unique copy of the irreducible
$\operatorname{Spin}(V)$-submodule $E\oplus E^{\vee}$.

######  Remark 2.2.

By uniqueness, the map $\Phi$ is intrinsically related to the skew-apolarity
map defined in [Arr+21, Definition 4]. Consider the splitting in
$\operatorname{SL}(E)$-modules

$\bigwedge^{ev}E\otimes\bigwedge^{od}E^{\vee}=\bigoplus_{r=0}^{\frac{N}{2}}\bigoplus_{s=0}^{\frac{N-2}{2}}\left(\bigwedge^{2r}E\otimes\bigwedge^{2s+1}E^{\vee}\right)\
.$

The map $\Phi$ is $\operatorname{Spin}(V)$-equivariant, hence
$\operatorname{SL}(E)$-equivariant, so are its restrictions

$\Phi_{2r,2s+1}:\bigwedge^{2r}E\otimes\bigwedge^{2s+1}E^{\vee}\longrightarrow
E\oplus E^{\vee}\ .$

By Schur’s lemma, each restriction $\Phi_{2r,2s+1}$ is non-zero if and only if
either $E$ or $E^{\vee}$ appears as irreducible $\operatorname{SL}(E)$-summand
in the tensor product on the left-hand side: this happens if and only if
$|2r-(2s+1)|=1$. In particular, it holds
$\operatorname{Im}\left(\Phi_{2r,2s+1}\right)=E$ if $2r=(2s+1)+1$,
$\operatorname{Im}\left(\Phi_{2r,2s+1}\right)=E^{\vee}$ if $2r=(2s+1)-1$, and
$\operatorname{Im}\left(\Phi_{2r,2s+1}\right)=0$ otherwise.
For $2r=(2s+1)+1$ (resp. $2r=(2s+1)-1$), the module
$\bigwedge^{2r}E\otimes\bigwedge^{2s+1}E^{\vee}$ has a unique copy of $E$
(resp. $E^{\vee}$) as irreducible $\operatorname{SL}(E)$-submodule, thus (up
to scalars) there exists a unique $\operatorname{SL}(E)$-equivariant morphism
from the tensor product onto $E$ (resp. $E^{\vee}$). By uniqueness, the
restrictions $\Phi_{2r,2s+1}$ are given (up to composing with a projection
$\pi_{E\oplus E^{\vee}}$ onto $E\oplus E^{\vee}$) by generalizing the skew-
catalecticant maps $\mathcal{C}^{s,d-s}_{t}$ in [Arr+21, Definition 4] as
follows:

$\begin{matrix}\Phi_{2r,2s+1}:&\bigwedge^{2r}E\otimes\bigwedge^{2s+1}E^{\vee}&\rightarrow&\bigwedge^{2r-(2s+1)}E\oplus\bigwedge^{2s+1-2r}E^{\vee}&\stackrel{{\scriptstyle\pi_{E\oplus
E^{\vee}}}}{{\twoheadrightarrow}}&E\oplus E^{\vee}\\\ &e\otimes
f&\mapsto&\mathcal{C}^{2s+1,2r-(2s+1)}_{e}(f)+\mathcal{C}^{2r,2s+1-2r}_{f}(e)\end{matrix}$

where $\mathcal{C}_{t}^{s,d-s}=0$ for $s>d$: roughly, depending on the sign of
$2r-(2s+1)$, one looks at vectors in $E^{\vee}$ as derivations on $E$, or
viceversa.

######  Definition 2.3.

The Clifford apolarity action is the $\operatorname{Spin}(E\oplus
E^{\vee})$-equivariant map

$\begin{matrix}\Phi:&\bigwedge^{ev}E\otimes\bigwedge^{od}E^{\vee}&\longrightarrow&E\oplus
E^{\vee}\\\ &e\otimes
f&\mapsto&C_{e}(f)_{|_{E}}+C_{f}(e)_{|_{E^{\vee}}}\end{matrix}$ (2.2)

where $C_{e}(f)_{|_{E}}$ (resp. $C_{f}(e)_{|_{E^{\vee}}}$) is the projection
onto $E$ (resp. $E^{\vee}$) of the contraction $f\cdot e\in\bigwedge E$ (resp.
$e\cdot f\in\bigwedge E^{\vee}$) obtained via Clifford multiplication.

#### Vanishing of sections in $H^{0}(\mathbb{S}_{N}^{+},\mathcal{U}(1))$.

Via the isomorphism
$H^{0}(\mathbb{S}_{N}^{+},\mathcal{U}(1))\simeq(V_{\omega_{N-1}}^{D_{N}})^{\vee}=\bigwedge^{od}E^{\vee}$,
any section $j_{f}\in H^{0}(\mathbb{S}_{N}^{+},\mathcal{U}(1))$ corresponds to
a spinor $f\in\bigwedge^{od}E^{\vee}$: we describe the zero locus of a section
$j_{f}$ by generalizing an argument from [Man21, proof of Prop. 2].
For any $[a]\in\mathbb{S}_{N}^{+}$, one has the fiber
$(\mathcal{U}(1))_{[a]}\simeq\operatorname{Hom}((\mathcal{O}(1))^{\vee}_{[a]},\mathcal{U}_{[a]})$.
In particular, it holds $(\mathcal{O}(1))^{\vee}_{[a]}=\langle
a\rangle_{\mathbb{C}}\subset\bigwedge^{ev}E$ and
$\mathcal{U}_{[a]}=H_{a}:=\\{v\in E\oplus E^{\vee}\ |\ v\cdot a=0\\}$. Thus we
can identify $j_{f}([a])\in(\mathcal{U}(1))_{[a]}$ with a homomorphism
$j_{f}([a]):\langle a\rangle_{\mathbb{C}}\rightarrow H_{a}$. We want to
determine $j_{f}([a])(\lambda a)\in H_{a}$ as the scalar
$\lambda\in\mathbb{C}$ varies.

######  Remark 2.4.

The scalar product $q(\cdot,\cdot)$ on $V=E\oplus E^{\vee}$ extends to a
scalar product $\langle\cdot,\cdot\rangle$ on $\bigwedge V$ defined on
decomposable elements $x=x_{1}\wedge\ldots\wedge x_{k}\in\bigwedge^{k}V$ and
$y=y_{1}\wedge\ldots\wedge y_{h}\in\bigwedge^{h}V$ as $\langle
x,y\rangle=\det(q(x_{i},y_{j}))$ if $k=h$, and zero otherwise. The isomorphism
of vector spaces $\bigwedge V\simeq\operatorname{Cl}(V,q)$ defines such scalar
product on the Clifford algebra too. The well-known adjointness between
exterior product and contraction $\langle v\wedge x,y\rangle=\langle x,v\neg
y\rangle$ for $v\in V$, $x\in\bigwedge^{h-1}V$ and $y\in\bigwedge^{h}V$,
together with the Clifford multiplication (1.1), implies $\langle x,v\cdot
y\rangle=\langle v\cdot x,y\rangle$ for any $x,y\in\bigwedge V$ and any $v\in
V$: in the Clifford algebra, this property extends to any element
$z\in\operatorname{Cl}(V,q)$ as $\langle x,zy\rangle=\langle z^{*}x,y\rangle$,
where $z^{*}$ is as in (1.2).

The function $\psi_{f,a}(v)=\langle f,v\cdot a\rangle$ is linear in $v\in V$
and vanishes on $H_{a}$, thus it belongs to
$H_{a}^{\perp}=\\{\psi:V\rightarrow\mathbb{C}\ |\ \psi(H_{a})=0\\}$. But
$H_{a}=H_{a}^{\perp}$ (since $a$ is a pure spinor), thus
$\psi_{f,\bullet}\in\operatorname{Hom}(\langle a\rangle_{\mathbb{C}},H_{a})$.
It follows that we have two maps

$\begin{matrix}&\bigwedge^{od}E^{\vee}&\longrightarrow&H^{0}\left(\mathbb{S}_{N}^{+},\mathcal{U}(1)\right)\\\
j:&f&\mapsto&j_{f}\\\ \Psi:&f&\mapsto&\psi_{f,\bullet}\end{matrix}$

where $j$ is the $\operatorname{Spin}(V)$-equivariant isomorphism given by
Borel–Weil Theorem. Next remark shows that the maps $j$ and $\Psi$ coincide up
to scalars.

######  Remark 2.5.

The map $\Psi$ is $\operatorname{Spin}(V)$-equivariant too. Indeed, for any
$G$-homogeneous bundle on a variety $G/P$, the $G$-action on a global section
$s$ is given by $(g\cdot s)(a)=s(g^{-1}\cdot a)$. Since for any
$x\in\operatorname{Spin}(V)$ it holds $x^{-1}=x^{*}$, one gets
$\left(x\cdot\psi_{f,\bullet}\right)(a)=\psi_{f,\bullet}(x^{*}a)=\psi_{f,x^{*}a}\in\operatorname{Hom}\left(\langle
x^{*}a\rangle,H_{x^{*}a}\right)$ where $\psi_{f,x^{*}a}(\lambda x^{*}a)\in
H_{x*a}^{\perp}=H_{x^{*}a}$ is the functional on $V$ given by
$\psi_{f,x^{*}a}(\lambda x^{*}a)(v)=\langle f,v\cdot\lambda x^{*}a\rangle$.
But $x^{*}\in\operatorname{Spin}(V)$ acts by conjugacy on $V$, thus
$V=x^{*}Vx$ and by adjointness one concludes $\langle f,v\cdot\lambda
x^{*}a\rangle=\langle f,(x^{*}wx)\cdot\lambda x^{*}a\rangle=\langle
f,x^{*}w\cdot\lambda a\rangle\stackrel{{\scriptstyle adj.}}{{=}}\langle x\cdot
f,w\cdot\lambda a\rangle$, that is $x\cdot\psi_{f,\bullet}=\psi_{x\cdot
f,\bullet}$.

######  Lemma 2.6.

In the previous notation, the zero locus $\mathcal{Z}(j_{f})$ of a global
section $j_{f}\in H^{0}(\mathbb{S}_{N}^{+},\mathcal{U}(1))$ corresponding to a
spinor $f\in\bigwedge^{od}E^{\vee}$ is

$\mathcal{Z}(j_{f})=\mathbb{S}_{N}^{+}\cap(V\cdot f)^{\perp}\ .$ (2.3)

###### Proof.

By Schur’s lemma, the equivariant maps $j$ and $\Psi$ coincide up to scalar,
hence the section $j_{f}\in H^{0}(\mathbb{S}_{N}^{+},\mathcal{U}(1))$ is such
that $j_{f}([a])(\lambda a)=\langle f,\bullet\cdot(\lambda a)\rangle\in H_{a}$
for any $\lambda\in\mathbb{C}$. We conclude that the zero locus
$\mathcal{Z}(j_{f})$ of the global section $j_{f}$ is given by the pure
spinors $[a]\in\mathbb{S}_{N}^{+}$ such that $0=\langle f,v\cdot
a\rangle=\langle v\cdot f,a\rangle$ for any $v\in V$, that is the thesis. ∎

##  3 The poset of $\operatorname{Spin}_{2N}$–orbits in
$\sigma_{2}(\mathbb{S}_{N}^{+})$

We assume the setting in Remark 1.1. The secant variety of lines
$\sigma_{2}(\mathbb{S}^{+}_{N})$ is invariant under
$\operatorname{Spin}_{2N}$–action, as well as its subsets
$\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$ and $\tau(\mathbb{S}_{N}^{+})$. We
refer to the orbits lying in $\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$ as
secant orbits, and to the orbits lying in $\tau(\mathbb{S}_{N}^{+})$ as
tangent orbits. Recall that the Spinor variety has diameter
$\operatorname{diam}(\mathbb{S}_{N}^{+})=\frac{N}{2}$. Moreover, for any
$l=1:\left(\frac{N}{2}-1\right)$ we consider the following pure spinors and
their corresponding maximal fully isotropic subspaces

$\begin{matrix}\mathbb{e}_{[N-2l]}\ =\ e_{1}\wedge\ldots\wedge e_{N-2l}\
\in\mathbb{S}_{N}^{+}\ ,\\\ \\\ E_{N-2l}:=H_{\mathbb{e}_{[N-2l]}}=\langle
e_{1},\ldots,e_{N-2l},f_{N-2l+1},\ldots,f_{N}\rangle_{\mathbb{C}}\
\in\operatorname{OGr}^{+}(N,V)\ .\end{matrix}$ (3.1)

By convention, for $l=\frac{N}{2}$ we set
$\mathbb{e}_{[0]}=\ell_{\omega_{N}}=\mathbbm{1}$. On the other hand, for $l=0$
one gets $\mathbb{e}_{[N]}=v_{\omega_{N}}$.

####  $Spin_{2N}$–orbits in $\mathbb{S}_{N}^{+}\times\mathbb{S}_{N}^{+}$.

The action of $\operatorname{Spin}_{2N}$ on $\mathbb{S}_{N}^{+}$ naturally
induces the action on the direct product
$\mathbb{S}_{N}^{+}\times\mathbb{S}_{N}^{+}$ given by
$g\cdot([a],[b])=([g\cdot a],[g\cdot b])$. We recall that the notation $g\cdot
a$ stands for the action of
$\operatorname{Spin}^{q}(V)\subset(\operatorname{Cl}_{q}(V))^{\times}$ on
$\bigwedge E$ via Clifford multiplication (1.1).

######  Remark 3.1.

Via the bijection (1.4), the action on
$\mathbb{S}_{N}^{+}\times\mathbb{S}_{N}^{+}$ is equivalent to the action of
$\operatorname{Spin}^{q}(V)$ on $\operatorname{OGr}^{+}(N,V)^{\times 2}$ given
by $g\cdot(H_{a},H_{b})=(g\cdot H_{a},g\cdot H_{b})$, where $g\cdot
H_{a}=gH_{a}g^{-1}$ is the conjugacy action: this follows from the inclusion
$\operatorname{Spin}^{q}(V)\subset N_{\operatorname{Cl}_{q}(V)^{\times}}(V)$.

From Remark 1.3, the Hamming distance in $\mathbb{S}_{N}^{+}$ is invariant
under $\operatorname{Spin}^{q}(V)$-action, as well as the dimensions of
subspaces in $V$ are preserved under conjugacy. In particular, the actions of
$\operatorname{Spin}^{q}(V)$ on $\mathbb{S}_{N}^{+}\times\mathbb{S}_{N}^{+}$
and on $\operatorname{OGr}^{+}(N,V)^{\times 2}$ restrict to actions on the
subsets

$\mathcal{O}_{l,N}:=\left\\{([a],[b])\in\mathbb{S}_{N}^{+}\times\mathbb{S}_{N}^{+}\
|\ d([a],[b])=l\right\\}$ (3.2)
$\widehat{\mathcal{O}}_{l,N}:=\left\\{(H_{a},H_{b})\in\operatorname{OGr}^{+}(N,V)^{\times
2}\ |\ \dim(H_{a}\cap H_{b})=N-2l\right\\}$ (3.3)

respectively, for any $l=0:\frac{N}{2}$ where
$\frac{N}{2}=\operatorname{diam}(\mathbb{S}_{N}^{+})$. Since for $l=0$ one
gets the diagonals $\Delta_{\mathbb{S}_{N}^{+}}$ and
$\Delta_{\operatorname{OGr}^{+}(N,V)}$ we consider $l\geq 1$. From Proposition
1.4, for any distance $l\geq 1$, the subsets $\mathcal{O}_{l,N}$ and
$\widehat{\mathcal{O}}_{l,N}$ are equivalent one to the other, and they give
partitions of the corresponding direct products (as sets). Since
$\dim(H_{\mathbb{e}_{[N]}}\cap H_{\mathbbm{e}_{l}})=\dim(E\cap E_{l})=N-2l$,
for any $l=1:\frac{N}{2}$ one gets the inclusion
$\operatorname{Spin}^{q}(V)\cdot\left([\mathbb{e}_{[N]}],[\mathbb{e}_{[N-2l]}]\right)\subset\mathcal{O}_{l,N}$.

######  Remark.

It is likely well-known that the $\operatorname{Spin}_{2N}$–orbit partition of
$\operatorname{Q}^{2N-2}\times\operatorname{Q}^{2N-2}$ is uniquely determined
by the Hamming distance: in particular, given
$\Delta_{\operatorname{Q}^{2N-2}}$ the diagonal, it holds

$\operatorname{Q}^{2N-2}\times\operatorname{Q}^{2N-2}\ =\
\Delta_{\operatorname{Q}^{2N-2}}\ \sqcup\
\operatorname{Spin}_{2N}\cdot([e_{1}],[e_{2}])\ \sqcup\
\operatorname{Spin}_{2N}\cdot([e_{1}],[f_{1}])\ .$ (3.4)

######  Proposition 3.2.

For any $l=1:\operatorname{diam}(\mathbb{S}_{N}^{+})$, the spin group
$\operatorname{Spin}^{q}(V)$ acts transitively on
$\mathcal{O}_{l,N}\subset\mathbb{S}_{N}^{+}\times\mathbb{S}_{N}^{+}$. In
particular, it holds
$\mathcal{O}_{l,N}=\operatorname{Spin}^{q}(V)\cdot([\mathbb{e}_{[N]}],[\mathbb{e}_{[N-2l]}])$.

###### Proof.

Given $([a],[b])\in\mathcal{O}_{l,N}$, by homogeneity of $\mathbb{S}_{N}^{+}$
we may assume $a=\mathbb{e}_{[N]}$.
First we prove the result for $l=\frac{N}{2}$. Since
$d([\mathbb{e}_{[N]}],[b])=\frac{N}{2}$, it holds $E\cap H_{b}=\\{0\\}$ and we
may assume $H_{b}=\langle g_{1},\ldots,g_{N}\rangle_{\mathbb{C}}$ for
generators $g_{j}$ as in (1.5). In light of (3.4) $\operatorname{Spin}^{q}(V)$
conjugates the fully isotropic vector
$g_{1}=f_{1}+\sum_{i=2}^{N}\alpha_{i1}e_{i}$ (having Hamming distance $2$ from
$e_{1}$) to $f_{1}$ by leaving $e_{1}$ fixed. Now, consider
$V^{\prime}=\langle e_{2},\ldots,e_{N},f_{2},\ldots,f_{N}\rangle_{\mathbb{C}}$
and the subspaces $E^{\prime}=E\cap V^{\prime}$ and
$H_{b}^{\prime}=H_{b}/\langle e_{1},f_{1}\rangle_{\mathbb{C}}$: again
$\operatorname{Spin}(V^{\prime})$ conjugates $g_{2}-\alpha_{12}e_{1}\in
H_{b}^{\prime}$ to $f_{2}$ by leaving $e_{2}$ fixed. In particular,
$\operatorname{Spin}^{q}(V)$ conjugates $g_{2}\in H_{b}$ to $f_{2}$ by leaving
$e_{1},f_{1},e_{2}$ fixed. By iterating, $\operatorname{Spin}^{q}(V)$
conjugates $H_{b}$ to $E^{\vee}$, hence $([\mathbb{e}_{[N]}],[b])$ to
$(\mathbb{e}_{[N]},[\mathbbm{1}])$.
On the other hand, for $l\lneq\frac{N}{2}$ one has $H_{b}=\langle
h_{1},\ldots,h_{N-2l},g_{N-2l+1},\ldots,g_{N}\rangle_{\mathbb{C}}$ for
$g_{j}$’s as in (1.5) and $E\cap H_{b}=\langle
h_{1},\ldots,h_{N-2l}\rangle_{\mathbb{C}}$. Up to reordering
$h_{1},\ldots,h_{N-2l}$, one gets that $h_{j}$ has Hamming distance $1$ from
$e_{j}$ for any $j=1:N-2l$, hence from (3.4) and similar arguments as above
one can conjugates $E\cap H_{b}$ to $E_{N-2l}$. Finally, by working in
$W:=V/E_{N-2l}$ one can conjugates $E/E_{N-2l}$ to $H_{b}/E_{N-2l}$ via
$\operatorname{Spin}(W)$ and lifting this to a conjugation under
$\operatorname{Spin}^{q}(V)$ leaving $E_{N-2l}$ fixed. The thesis follows. ∎

######  Corollary 3.3.

The product $\mathbb{S}_{N}^{+}\times\mathbb{S}_{N}^{+}$ splits in the
$\operatorname{Spin}_{2N}$–orbits

$\mathbb{S}_{N}^{+}\times\mathbb{S}_{N}^{+}=\Delta_{\mathbb{S}_{N}^{+}}\sqcup\bigsqcup_{l=1}^{\frac{N}{2}}\operatorname{Spin}_{2N}\cdot\left([\mathbb{e}_{[N]}],[\mathbb{e}_{[N-2l]}]\right)\
.$

###  3.1 Secant orbits in $\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$

We deduce the orbit partition of the dense subset
$\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$ from Proposition 3.2. We recall that
each spinor $[a+b]\in\mathbb{P}\left(\bigwedge^{ev}E\right)$ defines, via the
map (1.3), a fully isotropic subspace
$H_{a+b}=\operatorname{ker}(\psi_{a+b})$, which has maximal dimension $N$ if
and only if $[a+b]$ is pure. By definition of these subspaces as annihilators,
it clearly holds $H_{a}\cap H_{b}\subseteq H_{a+b}$.
For any two distinct pure spinors $[a],[b]\in\mathbb{S}_{N}^{+}$ such that
$d([a],[b])=1$, the spinor $[a+b]$ is pure too, since the line $L([a],[b])$
fully lies in the Spinor variety: in particular, in this case one has
$\dim(H_{a}\cap H_{b})\stackrel{{\scriptstyle\eqref{prop:hamming
distance}}}{{=}}N-2$ while $\dim H_{a+b}=N$, thus the strict inclusion
$H_{a}\cap H_{b}\subsetneq H_{a+b}$ holds. However, for higher Hamming
distances the equality holds.

######  Lemma 3.4.

In the above notation, it holds
$H_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}}=E\cap E_{N-2l}$ if and only if
$l\neq 1$. In particular, for any $l\geq 2$, it holds $\dim
H_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}}=N-2l$.

###### Proof.

We already know that $\langle e_{1},\ldots,e_{N-2l}\rangle_{\mathbb{C}}=E\cap
E_{N-2l}\subset H_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}}$ holds. Assume that
there exists $v\in H_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}}$ such that
$v\notin(E\cap E_{N-2l})\setminus\\{0\\}$. Thus, since
$H_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}}$ is fully isotropic, up to linear
combinations we can consider a decomposition of $v$ with respect to the
standard hyperbolic basis $(e_{1},\ldots,e_{N},f_{1},\ldots,f_{N})$ of the
form
$v=\alpha_{1}e_{N-2l+1}+\ldots+\alpha_{2l}e_{N}+\beta_{1}f_{N-2l+1}+\ldots+\beta_{2l}f_{N}$
for some $\alpha_{i},\beta_{j}\in\mathbb{C}$. Since
$H_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}}=\operatorname{ker}(\psi_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}})$,
it holds

$\displaystyle 0$ $\displaystyle=v\cdot(\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]})$
$\displaystyle\stackrel{{\scriptstyle\eqref{multiplication}}}{{=}}\sum_{i=1}^{2l}\alpha_{i}(e_{1}\wedge\ldots\wedge
e_{N-2l}\wedge
e_{N-2l+i})+\sum_{j=1}^{2l}(-1)^{j+1}\beta_{j}(e_{1}\wedge\ldots\wedge\hat{e}_{N-2l+j}\wedge\ldots\wedge
e_{N})\ ,$

where $\hat{e}_{N-2l+j}$ denotes that the vector is missing in the wedge
product.
Now, the summands above may simplify one to each other if and only if $l=1$,
otherwise they do not since they are all independent vectors in the standard
basis of $\bigwedge E$. It follows that, for $l\geq 2$, it holds
$v\cdot(\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]})=0$ if and only if
$\alpha_{i}=\beta_{j}=0$ for all $i,j\in[2l]$, that is
$H_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}}=E\cap E_{l}$. On the other hand, for
$l=1$, the conditions $\alpha_{1}=\beta_{2}$ and $\alpha_{2}=-\beta_{1}$ give
a non-zero vector $0\neq v\in
H_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}}\setminus(E\cap E_{l})$, thus $E\cap
E_{1}\subsetneq H_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}}=\langle
e_{1},\ldots,e_{N-2},e_{N-1}+f_{N},e_{N}-f_{N-1}\rangle_{\mathbb{C}}$. ∎

######  Corollary 3.5.

For any
$[a+b]\in\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})\setminus\mathbb{S}_{N}^{+}$,
the equality $H_{a+b}=H_{a}\cap H_{b}$ holds.

###### Proof.

Let
$[a+b]=[c+d]\in\sigma^{\circ}(\mathbb{S}_{N}^{+})\setminus\mathbb{S}_{N}^{+}$
be such that $d([a],[b])=l$ and $d([c],[d])=m$ for certain $2\leq
l,m\leq\frac{N}{2}$. By Proposition 3.2 there exists
$g\in\operatorname{Spin}^{q}(V)$ such that
$g\cdot([a],[b])=([\mathbb{e}_{[N]}],[\mathbb{e}_{[N-2l]}])$ and
$g\cdot([c],[d])=([c^{\prime}],[d^{\prime}])$. In particular,
$[\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}]=g\cdot[a+b]=g\cdot[c+d]=[c^{\prime}+d^{\prime}]$,
and by Proposition 3.4 we get $H_{c^{\prime}}\cap H_{d^{\prime}}\subset
H_{c^{\prime}+d^{\prime}}=H_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}}=E\cap
E_{N-2l}$, where the last equality follows from Lemma 3.4 since $l\geq 2$.
Dimensionally we have $N-2m=\dim(H_{c^{\prime}}\cap
H_{d^{\prime}})\leq\dim(E\cap E_{N-2l})=N-2l$. But, by symmetry, one also gets
$N-2l=\dim(H_{a^{\prime}}\cap H_{b^{\prime}})\leq\dim(E\cap E_{N-2m})=N-2m$,
thus $l=m$ and the thesis follows. ∎

Corollary 3.5 allows to define for any $l=2:\frac{N}{2}$ the subset

$\displaystyle\Sigma_{l,N}$
$\displaystyle:=\left\\{[a+b]\in\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})\ |\
\dim H_{a+b}=N-2l\right\\}$ $\displaystyle\
=\left\\{[a+b]\in\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})\ |\
d([a],[b])=l\right\\}\ .$ (3.5)

Moreover, we set $\Sigma_{1,N}:=\mathbb{S}_{N}^{+}$. Our claim is that the
subsets $\Sigma_{l,N}$ are exactly the $\operatorname{Spin}^{q}(V)$-orbits in
$\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$.
The action of $\operatorname{Spin}^{q}(V)$ on $\sigma_{2}(\mathbb{S}_{N}^{+})$
preserves the subsets $\Sigma_{l,N}$, as by Remark 3.1 the spin group acts on
$V$ and its subspaces by conjugacy. Moreover, by Proposition 3.2 any two pairs
$([a],[b])$ and $([c],[d])$ of Hamming distance $l$ are conjugated, hence
their lines $L([a],[b])$ and $L([c],[d])$ are so. Finally, the following
result proves that $\operatorname{Spin}^{q}(V)$ acts transitively on points on
a same line $L([a],[b])\setminus\\{[a],[b]\\}$ too.

######  Lemma 3.6.

For any two distinct pure spinors $[a],[b]\in\mathbb{S}_{N}^{+}$, the spin
group $\operatorname{Spin}^{q}(V)$ acts transitively on
$L([a],[b])\setminus\\{[a],[b]\\}$.

###### Proof.

Since the lines defined by pairs of pure spinors having the same Hamming
distance are all conjugated, it is enough to prove the transitivity on the
line $L([\mathbb{e}_{[N]}],[\mathbb{e}_{[N-2l]}])$. Moreover, given a point
$[\lambda\mathbb{e}_{[N]}+\mu\mathbb{e}_{[N-2l]}]=[\mathbb{e}_{[N]}+z\mathbb{e}_{[N-2l]}]\in
L([\mathbb{e}_{[N]}],[\mathbb{e}_{[N-2l]}])$, we can rewrite it as
$\mathbb{e}_{[N]}+z\mathbb{e}_{[N-2l]}=e_{1}\wedge\ldots\wedge
e_{N-2l}\wedge\left(e_{N-2l+1}\wedge\ldots\wedge e_{N}+z\mathbbm{1}\right)$.
Since $[e_{N-2l+1}\wedge\ldots\wedge e_{N}]$ and $[z\mathbbm{1}]$ are pure
spinors in $\mathbb{S}_{2l}^{+}$, we can restrict to consider the line
$L([\mathbb{e}_{[N]}],[\mathbbm{1}])$.
Given $\mathbb{e}_{[N]}+z\mathbbm{1}$, we look for a spin element
$g\in\operatorname{Spin}^{q}(V)$ such that
$\mathbb{e}_{[N]}+z\mathbbm{1}=k(\mathbb{e}_{[N]}+\mathbbm{1})$ for some
$k\in\mathbb{C}^{\times}$. We consider the element
$\tilde{g}=(a_{1}e_{1}+b_{1}f_{1})\cdots(a_{N}e_{N}+b_{N}f_{N})\in\operatorname{Cl}_{q}^{+}(V)$
for certain $a_{i},b_{i}\in\mathbb{C}^{\times}$: it is product of an even
number of vectors in $V$ by the assumption as $N$ is even. Then
$\tilde{g}\in\operatorname{Spin}^{q}(V)$ if and only if it is invertible and
it has unitary spinor norm, that is if and only if
$\prod_{i=1}^{N}a_{i}b_{i}\stackrel{{\scriptstyle\clubsuit}}{{=}}1$. Via
Clifford multiplication (1.1) it holds
$(a_{1}e_{1}+b_{1}f_{1})\cdot(\mathbb{e}_{[N]}+z\mathbbm{1})=a_{1}ze_{1}+b_{1}e_{2}\wedge\ldots\wedge
e_{N}$, and by iterating for $i=1:N$ one gets
$\tilde{g}\cdot(\mathbb{e}_{[N]}+z\mathbbm{1})=a_{1}\cdots
a_{N}z\mathbb{e}_{[N]}+b_{1}\cdots b_{N}\mathbbm{1}$. In particular, the
condition
$\tilde{g}\cdot(\mathbb{e}_{[N]}+z\mathbbm{1})=k(\mathbb{e}_{[N]}+\mathbbm{1})$
is equivalent to the condition
$z\prod_{i=1}^{N}a_{i}\stackrel{{\scriptstyle\spadesuit}}{{=}}\prod_{i=1}^{N}b_{i}$.
By putting together the conditions $(\clubsuit)$ and $(\spadesuit)$, it is
straightforward that for the choice $a_{1}=\ldots=a_{N}=\sqrt[2N]{z^{-1}}$ and
$b_{1}=\ldots=b_{N}=\sqrt[2N]{z}$ one gets
$\tilde{g}\cdot(\mathbb{e}_{[N]}+z\mathbbm{1})=\sqrt{z}(\mathbb{e}_{[N]}+\mathbbm{1})$.
∎

It follows that for any $l=1:\frac{N}{2}$ the subset $\Sigma_{l,N}$ is a
$\operatorname{Spin}^{q}(V)$-orbit in
$\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$. Moreover, by Proposition 3.2 we
deduce that for any $l$ it holds
$\Sigma_{l,N}=\operatorname{Spin}(V)\cdot[\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}]$.
In conclusion, we have proved the following theorem.

######  Theorem 3.7.

The dense set $\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$ splits under the action
of $\operatorname{Spin}_{2N}$ in the orbits

$\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})=\bigsqcup_{l=1}^{\frac{N}{2}}\operatorname{Spin}_{2N}\cdot[\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}]\
.$

###  3.2 Tangent orbits in $\tau(\mathbb{S}_{N}^{+})$

From the non-defectivity of $\sigma_{2}(\mathbb{S}_{N}^{+})$ and the dicotomy
between tangential and secant varieties [FH79, Corollary $4$], we know that
$\tau(\mathbb{S}_{4}^{+})=\sigma_{2}(\mathbb{S}_{4}^{+})=\mathbb{P}^{7}$ and
$\tau(\mathbb{S}_{5}^{+})=\sigma_{2}(\mathbb{S}_{5}^{+})=\mathbb{P}^{15}$,
while for $N\geq 6$ the tangential variety $\tau(\mathbb{S}_{N}^{+})$ is a
divisor in $\sigma_{2}(\mathbb{S}_{N}^{+})\subset\mathbb{P}^{2^{N-2}-1}$. We
deduce the orbit partition of $\tau(\mathbb{S}_{N}^{+})$ from the tangent
bundle on the Spinor variety.
Let $\mathcal{T}_{\mathbb{S}_{N}^{+}}$ be the tangent bundle on the Spinor
variety $\mathbb{S}_{N}^{+}$. From the parametrization of Spinor varieties,
one can describe the fiber at any pure spinor $[a]\in\mathbb{S}_{N}^{+}$ as

$\left(\mathcal{T}_{\mathbb{S}_{N}^{+}}\right)_{[a]}\ =\
T_{[a]}\mathbb{S}_{N}^{+}\ \simeq\ \bigwedge^{2}H_{a}\ ,$

leading to the isomorphism of homogeneous bundles
$\mathcal{T}_{\mathbb{S}_{N}^{+}}\ \simeq\ \bigwedge^{2}\mathcal{U}^{\vee}$,
where $\mathcal{U}$ is the rank–$N$ universal bundle on $\mathbb{S}_{N}^{+}$
obtained by pulling back the universal bundle on the Grassmannian
$\operatorname{Gr}(N,V)$.
By homogeneity of $\mathbb{S}_{N}^{+}$, all tangent spaces are conjugated one
to each other by transformations in $\operatorname{Spin}_{2N}\setminus
P_{\omega_{N}}$. Thus the $\operatorname{Spin}_{2N}$–orbits of points in the
tangential variety $\tau(\mathbb{S}_{N}^{+})$ are in bijection with the
$P_{\omega_{N}}$–orbits in the tangent space
$T_{[v_{\omega_{N}}]}\mathbb{S}_{N}^{+}\simeq\bigwedge^{2}E$, which are
parametrized by the possible ranks of skew-symmetric matrices of size $N$. Let
$\left[T_{[a]}\mathbb{S}_{N}^{+}\right]_{2l}$ be the set of tangent points to
$\mathbb{S}_{N}^{+}$ at the pure spinor $[a]$ corresponding to skew-symmetric
matrices of size $N$ and rank $2l$. Then for any $l=1:\frac{N}{2}$ we denote
the set of all tangent points corresponding to rank–$2l$ skew-symmetric
matrices by

$\Theta_{l,N}:=\left\\{[q]\in\tau(\mathbb{S}_{N}^{+})\ |\
\exists[a]\in\mathbb{S}_{N}^{+}\ :\
[q]\in\left[T_{[a]}\mathbb{S}_{N}^{+}\right]_{2l}\right\\}\ .$ (3.6)

The above arguments ensure that each subset $\Theta_{l,N}$ is indeed a
$\operatorname{Spin}^{q}(V)$-orbit, and all together they give the
$\operatorname{Spin}^{q}(V)$-orbit partition of $\tau(\mathbb{S}_{N}^{+})$.
Finally, for any $l=1:\frac{N}{2}$ the tangent orbit $\Theta_{l,N}$ admits as
representative the spinor

$[q_{l}]:=\left[\sum_{i=1}^{l}e_{2i-1}\wedge e_{2i}\right]\ .$ (3.7)

Indeed, the curve of rank–$2l$ skew-symmetric matrices of size $N$

$tC_{l}=t{\tiny\begin{bmatrix}P_{1}\\\ &\ddots\\\ &&P_{l}\\\
&&&0_{N-2l}\end{bmatrix}}\ \ \ \text{where}\ \ \
P_{i}={\scriptsize\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}}$ (3.8)

defines the curve of maximal fully isotropic subspaces

$H_{c(t)}=\left\langle
f_{1}+t\sum_{k=1}^{N}c_{k1}e_{k},\ldots,f_{N}+t\sum_{k=1}^{N}c_{kN}e_{k}\right\rangle_{\mathbb{C}}\
\in\ \operatorname{OGr}^{+}(N,V)\ ,$

which by Proposition 1.2 corresponds to the curve of pure spinors
$c(t)=\sum_{I\in 2^{[N]}}\operatorname{Pf}(C(t)_{I})\mathbb{e}_{I}$ passing at
$c(0)=\mathbbm{1}$ with direction $c^{\prime}(0)=q_{l}$, thus $[q_{l}]\in
T_{[\mathbbm{1}]}\mathbb{S}_{N}^{+}$. This proves the next theorem.

######  Theorem 3.8.

The tangential variety $\tau(\mathbb{S}_{N}^{+})$ splits in the
$\operatorname{Spin}_{2N}$-orbits

$\tau(\mathbb{S}_{N}^{+})\ =\
\bigsqcup_{l=1}^{\frac{N}{2}}\operatorname{Spin}_{2N}\cdot\left[\sum_{i=1}^{l}e_{2i-1}\wedge
e_{2i}\right]\ .$

###  3.3 Inclusions among closures of $\operatorname{Spin}_{2N}$–orbits

We have treated the secant orbits and the tangent orbits separately, now we
analyze the behaviour of their inclusions. First of all, notice that the
tangent representative $[q_{1}]=[e_{1}\wedge e_{2}]\in\Theta_{1,N}$ is a pure
spinor, hence

$\Theta_{1,N}=\mathbb{S}_{N}^{+}=\Sigma_{1,N}\ .$

######  Remark.

The orbit $\Theta_{1,N}$ is given by skew-symmetric matrices having rank (as
matrices) $2$, hence it is described by the Grassmannian of planes
$\operatorname{Gr}(2,N)$. This agrees with the more theoretical result [LM03,
Prop. 2.5 & Subsec. 3.1]
$T_{x}\mathbb{S}_{N}^{+}\cap\mathbb{S}_{N}^{+}\simeq\operatorname{Gr}(2,N)$
for any $x\in\mathbb{S}_{N}^{+}$.

Moreover, for $l=2$, a representative of $\Theta_{2,N}$ is
$[q_{2}]=[e_{1}\wedge e_{2}+e_{3}\wedge e_{4}]$. But $[e_{1}\wedge e_{2}]$ and
$[e_{3}\wedge e_{4}]$ are pure spinors with corresponding subspaces
$H_{e_{1}\wedge e_{2}}=E_{[2]}$ and $H_{e_{3}\wedge e_{4}}=\left\langle
f_{1},f_{2},e_{3},e_{4},f_{5},\ldots,f_{N}\right\rangle_{\mathbb{C}}$, hence
they have Hamming distance $2$ and $[q_{2}]\in\Sigma_{2,N}$. Thus

$\Theta_{2,N}=\Sigma_{2,N}\ .$

######  Remark 3.9.

Via the map $\psi_{q}$ in (1.3) every $[q]\in\Theta_{l,N}$ defines a subspace
$H_{q}\in\operatorname{OGr}(N-2l,V)$: indeed, the representative $[q_{l}]$
(3.7) defines the subspace $H_{q_{l}}=\langle
f_{2l+1},\ldots,f_{N}\rangle_{\mathbb{C}}$ of dimension $N-2l$ and
$\operatorname{Spin}^{q}(V)$ acts by conjugacy on the subspaces, preserving
their dimensions. Moreover, by definition, a tangent point $[q]\in
T_{[a]}\mathbb{S}_{N}^{+}\cap\Theta_{l,N}$ is the direction of a curve
$\gamma(t)=\\{[a(t)]\ |\ t\in(-\epsilon,\epsilon)\\}\subset\mathbb{S}_{N}^{+}$
passing at $\gamma(0)=[a]$ and, up to considering a smaller neighbourhood, one
can assume $d([a],[a(t)])=m$ for any $t\in(0,\epsilon)$. Then for any $t>0$
the spinor $\frac{a-a(t)}{t}$ defines a subspace
$H_{t}\in\operatorname{OGr}(N-2l,V)$ and one gets the equality
$H_{q}=\lim_{t\rightarrow 0}H_{t}\in\operatorname{OGr}(N-2l,V)$.

From the previous arguments we deduce the following description of the secant
variety of lines

$\sigma_{2}(\mathbb{S}_{N}^{+})=\mathbb{S}_{N}^{+}\sqcup\left(\bigsqcup_{l=2}^{\frac{N}{2}}\Sigma_{l,N}\right)\cup\left(\bigsqcup_{l=3}^{\frac{N}{2}}\Theta_{l,N}\right)\
,$

where the non-disjoint unions appears since we haven’t proved
$\Sigma_{l,N}\neq\Theta_{l,N}$ for $l\geq 3$ yet. Since we are interested in
considering $l\geq 3$, we assume $N\geq 6$.

######  Lemma 3.10.

  1. 1.

For any $l=2:\frac{N}{2}$ it holds
$\Sigma_{l-1,N}\subset\overline{\Sigma_{l,N}}$.

  2. 2.

For any $l=2:\frac{N}{2}$ it hols
$\Theta_{l-1,N}\subset\overline{\Theta_{l,N}}$.

  3. 3.

For any $l=3:\frac{N}{2}$ it holds
$\Theta_{l,N}\subset\overline{\Sigma_{l,N}}$.

###### Proof.

  1. 1.

For $\epsilon>0$ consider the sequence
$[\mathbb{e}_{[N]}+a_{\epsilon}]\in\Sigma_{l,N}$ for
$[a_{\epsilon}]\in\mathbb{S}_{N}^{+}$ defined by the subspaces
$H_{a_{\epsilon}}=\left\langle
e_{1},\ldots,e_{N-2l},g_{N-2l+1}(\epsilon),g_{N-2l+2}(\epsilon),g_{N-2l+3},\ldots,g_{N}\right\rangle_{\mathbb{C}}\in\operatorname{OGr}^{+}(N,V)$
where

$g_{N-2l+1}(\epsilon)=\frac{1}{\epsilon}f_{N-2l+1}+e_{N-2l+2}\ \ \ ,\ \ \
g_{N-2l+2}(\epsilon)=\frac{1}{\epsilon}f_{N-2l+2}-e_{N-2l+1}$

and $g_{h}=f_{h}+\sum_{k=N-2l+3}^{N}\alpha_{kj}e_{k}$ as in (1.5). Then the
sequence $[\mathbb{e}_{[N]}+a_{\epsilon}]$ has limit $[\mathbb{e}_{[N]}+a]$
where the pure spinor $[a]\in\mathbb{S}_{N}^{+}$ corresponds to the maximal
fully isotropic subspace $H_{a}=\left\langle
e_{1},\ldots,e_{N-2l+2},g_{N-2l+3},\ldots,g_{N}\right\rangle_{\mathbb{C}}$: in
particular, $[\mathbb{e}_{[N]}+a]\in\Sigma_{l-1,N}$. By reversing this
argument, one can always look at a point in $\Sigma_{l-1,N}$ as limit of a
sequence in $\Sigma_{l,N}$.

  2. 2.

The tangent points in $\Theta_{l-1,N}$ correspond to skew-symmetric matrices
of size $N$ and rank $2l-2$, while points in $\Theta_{l,N}$ to rank–$2l$ skew-
symmetric matrices of size $N$. As the former matrices lie in the closure of
the latter ones, the thesis follows.

  3. 3.

Consider $[q]\in\Theta_{l,N}\subset\tau(\mathbb{S}_{N}^{+})$: then by Remark
3.9 $\dim H_{q}=N-2l$ and there exists a curve of pure spinors
$\gamma(t)=\\{[a(t)]\ |\ t\in(-\epsilon,\epsilon)\\}\subset\mathbb{S}_{N}^{+}$
with direction $\gamma^{\prime}(0)=[q]$ such that (up to a smaller $\epsilon$)
$d([a(t)],[a(0)])=l$ for any $t\in(0,\epsilon)$. In particular, for any $t>0$
it holds $\left[\frac{a(t)+a(0)}{t}\right]\in\Sigma_{l,N}$ and by definition
$[q]=\left[\lim_{t\rightarrow 0}\frac{a(t)-a(0)}{t}\right]=\lim_{t\rightarrow
0}\left[\frac{a(t)+a(0)}{t}\right]\in\overline{\Sigma_{l,N}}$.

∎

######  Lemma 3.11.

For any $N\geq 6$ and $l=3:\frac{N}{2}$, it holds
$\Sigma_{l,N}\neq\Theta_{l,N}$.

###### Proof.

For $l=\frac{N}{2}$ the equality does not hold since for $N\geq 6$ one has
$\tau(\mathbb{S}_{N}^{+})\subsetneq\sigma_{2}(\mathbb{S}_{N}^{+})$.
By contradiction, assume that there exists $3\leq l\leq\frac{N}{2}-1$ such
that the equality holds. Then Lemma 3.10 implies $\Sigma_{i,N}=\Theta_{i,N}$
for any $3\leq i\leq l$: in particular, $\Sigma_{3,N}=\Theta_{3,N}$. Since we
want to deal with $3\lneq\frac{N}{2}$, we assume $N\geq 7$.
Consider the tangent representative $[q_{3}]=[e_{1}\wedge e_{2}+e_{3}\wedge
e_{4}+e_{5}\wedge e_{6}]\in\Theta_{3,N}$. It defines the subspace
$H_{q_{3}}=\langle f_{7},\ldots,f_{N}\rangle_{\mathbb{C}}$ of dimension $\dim
H_{q_{3}}=N-6$. Set $E\cap E_{[6]}=\langle
e_{1},e_{2},e_{3},e_{4},e_{5},e_{6}\rangle_{\mathbb{C}}$. Since
$[q_{3}]\in\Sigma_{3,N}$, there exist pure spinors
$[a],[b]\in\mathbb{S}_{N}^{+}$ such that $[q_{3}]=[a+b]$ and $d([a],[b])=3$.
Then, being kernels, one gets $\langle
f_{7},\ldots,f_{N}\rangle_{\mathbb{C}}=H_{q_{3}}=H_{a+b}=H_{a}\cap H_{b}$,
where the last equality follows from Corollary 3.5. This implies that
$a+b\in\bigwedge^{ev}(E\cap E_{[6]})$ and $[a],[b]\in\mathbb{S}_{6}^{+}$: in
particular, the equality $[q_{3}]=[a+b]$ holds in $\Sigma_{3,6}$ for the
representative $[q_{3}]\in\Theta_{3,6}$. But this means that
$\Theta_{3,6}=\Sigma_{3,6}$, that is
$\tau(\mathbb{S}_{6}^{+})=\sigma_{2}(\mathbb{S}_{6}^{+})$ which is a
contradiction. ∎

######  Theorem 3.12.

For any $N\geq 6$, the poset of $\operatorname{Spin}_{2N}$–orbits in the
secant variety of lines $\sigma_{2}(\mathbb{S}_{N}^{+})$ is described by the
graph in Figure (1), where arrows denote the inclusion of an orbit into the
closure of the other orbit. In particular, the orbits $\Theta_{\frac{N}{2},N}$
and $\Sigma_{\frac{N}{2},N}$ are the dense orbits of the tangential and secant
variety respectively.

$\mathbb{S}_{N}^{+}$$\Theta_{2,N}=\Sigma_{2,N}$$\Theta_{3,N}$$\vdots$$\Theta_{\frac{N}{2},N}$$\Sigma_{3,N}$$\vdots$$\Sigma_{\frac{N}{2},N}$
Figure 1: Poset graph of $\operatorname{Spin}_{2N}$–orbits in
$\sigma_{2}(\mathbb{S}_{N}^{+})$.

##  4 Identifiability in $\sigma_{2}(\mathbb{S}_{N}^{+})$

From Sec. 1 we recall the preliminary notions of (un)identifiability and
decomposition locus. We assume the setting in Remark 1.1.

#### Unidentifiability and decomposition loci for $\Sigma_{2,N}$

The $6$–dimensional Spinor variety $\mathbb{S}_{4}^{+}\subset\mathbb{P}^{7}$
coincides with the $6$-dimensional quadric
$\operatorname{Q}^{6}\subset\mathbb{P}^{7}$. In this case, the secant orbit
$\Sigma_{2,4}$ is dense in $\sigma_{2}(\mathbb{S}_{4}^{+})=\mathbb{P}^{7}$.
Given $[q]\in\mathbb{P}^{7}\setminus\operatorname{Q}^{6}$, for any pair
$([x],[y])\in Dec([q])$ in the decomposition locus it holds $d([x],[y])=2$,
since $L([x],[y])\nsubseteq\operatorname{Q}^{6}$.

######  Proposition 4.1.

For any $[q]\in\mathbb{P}^{7}\setminus\operatorname{Q}^{6}$, the decomposition
locus $Dec([q])$ is parametrized by the lines
$\mathbb{P}^{1}\subset\mathbb{P}^{7}$ passing at $[q]$ and intersecting
$\operatorname{Q}^{6}$ in two distinct points, i.e.

$Dec([q])\simeq\mathbb{P}^{6}\setminus\operatorname{Q}^{5}\ .$

In particular, the distance-$2$ orbit $\Sigma_{2,4}$ is unidentifiable.

###### Proof.

The set of lines $\mathbb{P}^{1}\subset\mathbb{P}^{7}$ passing through $[q]$
and intersecting $\operatorname{Q}^{6}$ in two distinct points is isomorphic
to $\mathbb{P}^{6}\setminus\operatorname{Q}^{5}$: indeed, points in
$\operatorname{Q}^{5}\subset\mathbb{P}^{6}$ correspond to lines
$\mathbb{P}^{1}\subset\mathbb{P}^{7}$ which are tangent to
$\operatorname{Q}^{6}$. A pair $([x],[y])\in(\operatorname{Q}^{6})^{\times 2}$
gives a decomposition for $[q]=[x+y]$ if and only if the line
$L([x],[y])\subset\mathbb{P}^{7}$ contains $[q]$ and intersects
$\operatorname{Q}^{6}$ exactly in the distinct points $[x],[y]$. Thus the
thesis follows. ∎

######  Example 4.2.

For $\mathbb{S}_{4}^{+}$, consider the representative
$[\mathbb{e}_{[4]}+\mathbbm{1}]$ of the dense orbit $\Sigma_{2,4}$: an
alternative decomposition is

$\left[\mathbb{e}_{[4]}+\mathbbm{1}\right]\ =\
[(\mathbb{e}_{[4]}-\mathbb{e}_{[2]})+(\mathbb{e}_{[2]}+\mathbbm{1})]$

where both summands are pure spinors since
$d([\mathbb{e}_{[4]}],[\mathbb{e}_{[2]}])=d([\mathbb{e}_{[2]}],[\mathbbm{1}])=1$.

######  Proposition 4.3.

For any $N\geq 6$, let $[q]\in\Sigma_{2,N}$ defining the $(N-4)$–dimensional
isotropic subspace $H_{q}\subset\mathbb{C}^{2N}$ via $\psi_{q}$ in (1.3). Let
$\operatorname{Q}^{6}\subset\mathbb{P}^{7}=\mathbb{P}(H_{q}^{\perp}/H_{q})$ be
the quadric in the (projectivization of the) orthogonal quotient of $H_{q}$.
For any $[\tilde{a}]\in\operatorname{Q}^{6}$, we denote by
$[a]=[\tilde{a}\wedge H_{q}]$ the pure spinor in $\mathbb{S}_{N}^{+}$. Then
the decomposition locus of $[q]$ is $6$–dimensional and isomorphic to the open
set

$Dec([q])\simeq\operatorname{Q}^{6}\setminus\left\\{[\tilde{a}]\in\operatorname{Q}^{6}\
|\ L([a],[q])\subset\mathbb{S}_{N}^{+}\right\\}\ .$

In particular, the distance-$2$ orbit $\Sigma_{2,N}$ is unidentifiable.

###### Proof.

Fix $[q]\in\Sigma_{2,N}$. For any $([a],[b])\in Dec([q])$, the pure spinors
$[a],[b]$ have Hamming distance $2$ and define two maximal fully isotropic
subspaces $H_{a},H_{b}$ such that $H_{a}\cap H_{b}=H_{q}$. In particular,
$\dim(H_{a}\cap H_{b})=N-4$ and $\dim\left(\frac{(H_{a}\cap
H_{b})^{\perp}}{H_{a}\cap H_{b}}\right)=8$. Thus, in the orthogonal quotient
space $W:=H_{q}^{\perp}/H_{q}\simeq\mathbb{C}^{8}$, they give the
$4$–dimensional (maximal) isotropic subspaces $H_{a}/H_{q},H_{b}/H_{q}$ which
intersect trivially, hence they correspond to two pure spinors
$[\tilde{a}],[\tilde{b}]\in\mathbb{S}_{4}^{+}\simeq\operatorname{Q}^{6}$ such
that $d([\tilde{a}],[\tilde{b}])=2$: in particular, the line
$L([\tilde{a}],[\tilde{b}])\nsubseteq\operatorname{Q}^{6}$ as well as
$L([a],[b])=L([a],[q])\nsubseteq\mathbb{S}_{N}^{+}$.
On the other hand, start from a pure spinor
$[\tilde{a}]\in\operatorname{Q}^{6}$ (look at the quadric as the Spinor
variety $\mathbb{S}_{4}^{+}$ constructed from $W$) and consider the lifting of
its corresponding subspace
$\tilde{H}_{\tilde{a}}\in\operatorname{OGr}^{+}(4,W)$ to the subspace
$H:=\langle\tilde{H}_{\tilde{a}},H_{q}\rangle_{\mathbb{C}}\in\operatorname{OGr}^{+}(N,V)$,
which corresponds to the pure spinor $[a]=[\tilde{a}\wedge
H_{q}]\in\mathbb{S}_{N}^{+}$. If $[\tilde{a}]\in\operatorname{Q}^{6}$ is such
that $L([a],[q])\nsubseteq\mathbb{S}_{N}^{+}$, then by Bézout
($\mathbb{S}_{N}^{+}$ is intersection of quadrics) there exists a unique
$[b]\in\mathbb{S}_{N}^{+}$ such that
$L([a],[q])\cap\mathbb{S}_{N}^{+}=\\{[a],[b]\\}$.
We conclude that each $[\tilde{a}]\in\operatorname{Q}^{6}$ such that
$L([a],[q])\nsubseteq\mathbb{S}_{N}^{+}$ corresponds to a unique pair
$([a],[b])\in Dec([q])$ in the decomposition locus of $[q]$, hence the thesis.
∎

#### Identifiability of orbits $\Sigma_{l,N}$ for $l\geq 3$.

We show the identifiability of the secant orbits $\Sigma_{l,N}$ for $l\geq 3$
via an inductive argument, based on the injectivity of a wedge-multiplication
map. The base case of the induction is given by the identifiability of the
dense orbit $\Sigma_{\frac{N}{2},N}$, which we prove via Clifford apolarity
introduced in Sec. 2.

######  Remark.

The case of the $15$–dimensional Spinor variety
$\mathbb{S}_{6}^{+}\subset\mathbb{P}^{31}$ was already known [AR03, Example 5]
as an example of variety with one apparent double point (OADP variety), namely
a $n$–dimensional variety $X\subset\mathbb{P}^{2n+1}$ such that through a
general point of $\mathbb{P}^{2n+1}$ there passes a unique secant line to $X$
[AR03, Definition 3].

######  Lemma 4.4.

For any $N\geq 6$, the dense secant orbit $\Sigma_{\frac{N}{2},N}$ is
identifiable.

###### Proof.

We prove that the representative
$[\mathbb{e}_{[N]}+\mathbbm{1}]\in\Sigma_{\frac{N}{2},N}$ is identifiable. In
the notation of Sec. 2, consider the map
$\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}}:H^{0}(\mathbb{S}_{N}^{+},\mathcal{U}(1))\rightarrow
H^{0}(\mathbb{S}_{N}^{+},\mathcal{U}^{\vee})^{\vee}$ corresponding to the
Clifford apolarity (2.2)

$\begin{matrix}\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}}:&\bigwedge^{od}E^{\vee}&\rightarrow&E\oplus
E^{\vee}\\\
&f&\mapsto&C_{\mathbb{e}_{[N]}+\mathbbm{1}}(f)_{|_{E}}+C_{f}(\mathbb{e}_{[N]}+\mathbbm{1})_{|_{E^{\vee}}}\end{matrix}\
.$

Since
$\operatorname{Rk}\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}}=2N=2\operatorname{Rk}\mathcal{U}(1)$,
from Proposition 1.8 it is enough to prove that the common zero locus of
$\operatorname{ker}(\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}})$ is
$\mathcal{Z}\left(\operatorname{ker}(\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}})\right)=\\{[\mathbb{e}_{[N]}],[\mathbbm{1}]\\}$.
Given
$f=\sum_{k=0}^{\frac{N-2}{2}}\sum_{I\in\binom{[N]}{2k+1}}c_{I}\mathbb{f}_{I}\in\bigwedge^{od}E^{\vee}$,
one gets

$\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}}(f)=\big{(}f\cdot\mathbb{e}_{[N]}+f\cdot\mathbbm{1}\big{)}_{|_{E}}+\big{(}\mathbb{e}_{[N]}\cdot
f+\mathbbm{1}\cdot
f\big{)}_{|_{E^{\vee}}}=\big{(}f\cdot\mathbb{e}_{[N]}\big{)}_{|_{E}}+f_{|_{E^{\vee}}}\
,$

hence

$\operatorname{ker}(\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}})=\bigwedge^{3}E^{\vee}\oplus\ldots\oplus\bigwedge^{N-3}E^{\vee}\
.$

Consider
$a=\sum_{s=0}^{\frac{N}{2}}\sum_{J\in\binom{[N]}{2s}}\alpha_{J}\mathbb{e}_{J}\in\bigwedge^{ev}E$
such that
$[a]\in\mathcal{Z}\left(\operatorname{ker}(\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}})\right)$.
From the vanishing of global sections in Lemma 2.6, we know that $\langle
V\cdot f,a\rangle=0$ for any
$f\in\operatorname{ker}(\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}})$. In particular,
for any index subset $I\subset[N]$ of odd cardinality from $3$ to $N-3$ (so
that $\mathbb{f}_{I}$ lies in the kernel), and for any basis vector
$e_{\lambda}\in E$, we get

$\displaystyle 0$ $\displaystyle=\langle
e_{\lambda}\cdot\mathbb{f}_{I},a\rangle=\langle\mathbb{f}_{I},e_{\lambda}\wedge
a\rangle=\sum_{s=0}^{\frac{N}{2}}\sum_{J\in\binom{[N]}{2s}}\alpha_{J}\langle\mathbb{f}_{I},e_{\lambda}\wedge\mathbb{e}_{J}\rangle$
(4.1) $\displaystyle=\sum_{s=0}^{\frac{N}{2}}\sum_{J\in\binom{[N]}{2s}\ :\
J\not\owns\lambda}(-1)^{pos(\lambda,J\cup\\{\lambda\\})+1}\alpha_{J}\langle\mathbb{f}_{I},\mathbb{e}_{J\cup\\{\lambda\\}}\rangle=(-1)^{pos(\lambda,I)+1}\alpha_{I\setminus\\{\lambda\\}}\
,$ (4.2)

where $pos(\lambda,I)$ denotes the position of $\lambda$ in the ordered subset
$I$, and the last equality follows from Remark 2.4. Thus in $a$ there is no
summand indexed by a subset $J$ such that $J\subset I$ and $|J|=|I|-1$ for any
subset $I$ with $|I|=3,5,\ldots,N-3$. On the other hand, for any subset
$I\subset[N]$ of odd cardinality from $3$ to $N-3$ (i.e.
$\mathbb{f}_{I}\in\operatorname{ker}(\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}})$)
and for any basis vector $f_{\nu}\in E^{\vee}$ such that $\nu\notin I$, it
holds

$\displaystyle 0$ $\displaystyle=\langle
f_{\nu}\cdot\mathbb{f}_{I},a\rangle=\langle
f_{\nu}\wedge\mathbb{f}_{I},a\rangle=(-1)^{pos(\nu,I\cup\\{\nu\\})+1}\sum_{s=0}^{\frac{n+1}{2}}\sum_{J\in\binom{[N]}{2s}}\alpha_{J}\langle\mathbb{f}_{I\cup\\{\nu\\}},\mathbb{e}_{J}\rangle$
(4.3) $\displaystyle=(-1)^{pos(\nu,I\cup\\{\nu\\})+1}\alpha_{I\cup\\{\nu\\}}\
,$ (4.4)

implying that $a$ has no summand indexed by a subset $J$ such that
$4\leq|J|\leq N-2$. We deduce that every
$[a]\in\mathcal{Z}\left(\operatorname{ker}(\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}})\right)$
is such that
$a=\alpha+\beta\mathbb{e}_{[N]}\in\mathbb{C}\oplus\bigwedge^{N}E$. Since
$d(\mathbbm{1},\mathbb{e}_{[N]})=\frac{N}{2}$, the line
$L([\mathbb{e}_{[N]}],[\mathbbm{1}])$ does not lie in $\mathbb{S}_{N}^{+}$: as
Spinor varieties are intersections of quadrics, by Bézout the only points of
the form $[\alpha\mathbbm{1}+\beta\mathbb{e}_{[N]}]$ being pure spinors are
for either $\alpha=0$ or $\beta=0$, that is
$\mathcal{Z}\left(\operatorname{ker}(\Phi_{\mathbb{e}_{[N]}+\mathbbm{1}})\right)=\\{[\mathbb{e}_{[N]}],[\mathbbm{1}]\\}$.
∎

######  Theorem 4.5.

For any $N\geq 6$ and $l\geq 3$, the secant orbit
$\Sigma_{l,N}\subset\sigma_{2}(\mathbb{S}_{N}^{+})$ is identifiable.

###### Proof.

Fix $l\geq 3$ and consider the orbit
$\Sigma_{l,N}=\operatorname{Spin}_{2N}\cdot[\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}]$.
By homogeneity, it is enough to show that the spinor
$[\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}]$ is identifiable.
Assume ad absurdum that there exist two pure spinors
$[a],[b]\in\mathbb{S}_{N}^{+}\setminus\\{[\mathbb{e}_{[N]}],[\mathbb{e}_{[N-2l]}]\\}$
such that $[a+b]=[\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}]$. In particular, since
$l\neq 1$, Corollary 3.5 implies that $H_{a}\cap
H_{b}=H_{a+b}=H_{\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}}=E\cap E_{N-2l}$, where
we recall that $E_{N-2l}=\langle
e_{1},\ldots,e_{N-2l},f_{N-2l+1},\ldots,f_{N}\rangle_{\mathbb{C}}$. Since
$\langle e_{1},\ldots,e_{N-2l}\rangle_{\mathbb{C}}=E\cap E_{N-2l}\subset
H_{a},H_{b}$, by Proposition 1.2 it follows that there exist two spinors
$a^{\prime},b^{\prime}\in\bigwedge^{ev}\left(\langle
e_{N-2l+1},\ldots,e_{N}\rangle_{\mathbb{C}}\right)$ such that
$a=\mathbb{e}_{[N-2l]}\wedge a^{\prime}$ and $b=\mathbb{e}_{[N-2l]}\wedge
b^{\prime}$: by maximality of $H_{a}$ and $H_{b}$, the spinors
$[a^{\prime}],[b^{\prime}]\in\mathbb{S}_{2l}^{+}$ are pure in a smaller Spinor
variety where they have maximum Hamming distance $l$. Moreover, since
$[a],[b]\in\mathbb{S}_{N}^{+}\setminus\\{[\mathbb{e}_{[N]}],[\mathbb{e}_{[N-2l]}]\\}$,
we also know that
$[a^{\prime}],[b^{\prime}]\in\mathbb{S}_{2l}^{+}\setminus\\{[e_{N-2l+1}\wedge\ldots\wedge
e_{N}],[\mathbbm{1}]\\}$.
The spinor
$[a+b]=[\mathbb{e}_{[N-2l]}\wedge(a^{\prime}+b^{\prime})]\in\Sigma_{l,N}$ is
the image of $[a^{\prime}+b^{\prime}]\in\Sigma_{l,2l}$ via the wedge-
multiplication map

$\mathbb{e}_{[N-2l]}\wedge\bullet\ :\ \bigwedge^{ev}\langle
e_{N-2l+1},\ldots,e_{N}\rangle_{\mathbb{C}}\ \longrightarrow\bigwedge^{ev}E$
(4.5)

restricting to
$(\mathbb{e}_{[N-2l]}\wedge\bullet):\Sigma_{l,2l}\rightarrow\Sigma_{l,N}$.
Since the above linear map is injective, the equality
$\big{(}\mathbb{e}_{[N-2l]}\wedge\bullet\big{)}([a^{\prime}+b^{\prime}])=\big{(}\mathbb{e}_{[N-2l]}\wedge\bullet\big{)}([e_{N-2l+1}\wedge\ldots\wedge
e_{N}+\mathbbm{1}])$ implies that
$[a^{\prime}+b^{\prime}]=[e_{N-2l+1}\wedge\ldots\wedge e_{N}+\mathbbm{1}]$.
But the spinor $[e_{N-2l+1}\wedge\ldots\wedge e_{N}+\mathbbm{1}]$ is a
representative for the dense orbit
$\Sigma_{l,2l}\subset\sigma_{2}(\mathbb{S}_{2l}^{+})$, which is identifiable
by Lemma 4.4 (since $l\geq 3$). Thus it holds
$\\{[a^{\prime}],[b^{\prime}]\\}=\\{[e_{N-2l+1}\wedge\ldots\wedge
e_{N}],[\mathbbm{1}]\\}$, in contradiction to
$[a^{\prime}],[b^{\prime}]\in\mathbb{S}_{2l}^{+}\setminus\\{[e_{N-2l+1}\wedge\ldots\wedge
e_{N}],[\mathbbm{1}]\\}$. ∎

##  5 Tangential-identifiability in $\tau(\mathbb{S}_{N}^{+})$

In this section we prove that any point of a tangent orbit $\Theta_{l,N}$ for
$l\geq 3$ is tangential-identifiable (cf. Definition 1.5), and we do so via
Clifford apolarity (cf. Section 2). We assume the setting in Remark 1.1. Since
for $N\leq 5$ it holds
$\overline{\Theta_{2,N}}=\tau(\mathbb{S}_{N}^{+})=\sigma_{2}(\mathbb{S}_{N}^{+})$,
in the following we assume $N\geq 6$.

######  Remark 5.1.

The $15$–dimensional Spinor variety $\mathbb{S}_{6}^{+}\subset\mathbb{P}^{31}$
has been deeply studied in the context of the Freudenthal’s magic square
[LM01]. Indeed, $\mathbb{S}_{6}^{+}$ is the third of the four Legendrian
varieties lying in the third row of the square (associated to the group
$\operatorname{Spin}_{12}$ \- see [LM01]), while
$\mathbb{P}^{31}=\sigma_{2}(\mathbb{S}_{6})^{+}$ is the third prehomogenous
space in the fourth row (associated to the exceptional group $E_{7}$ \- see
[Cle03]). In both the above references the generic tangential-identifiability
of $\tau(\mathbb{S}_{6}^{+})$ is obtained: we refer to [LM01, Propp. 5.8-5.12]
and [Cle03, Propp. 8.4,9.8] for details.

In the notation of Section 2, for a tangent spinor
$[q]\in\tau(\mathbb{S}_{N}^{+})\setminus\mathbb{S}_{N}^{+}$ we consider the
map

$\Phi_{q}:H^{0}\left(\mathbb{S}_{N}^{+},\mathcal{U}(1)\right)\longrightarrow
H^{0}\left(\mathbb{S}_{N}^{+},\mathcal{U}^{\vee}\right)^{\vee}$

corresponding to the Clifford apolarity in (2.2). Similarly to the proof of
Lemma 4.4, we apply the nonabelian apolarity, but in this case we deal with
non-reduced subschemes. Let $Y\subset\mathbb{S}_{N}^{+}$ be a non-reduced
subscheme of $\mathbb{S}_{N}^{+}$ of length $2$ such that $[q]\in\langle
Y\rangle$, and let $Y_{supp}$ be its support: in particular, such a $Y$
corresponds to $\\{[p],[q]\\}$ for $[p]\in\mathbb{S}_{N}^{+}$ such that
$[q]\in T_{[p]}\mathbb{S}_{N}^{+}$, and $Y_{supp}=\\{[p]\\}$. Then by
Proposition 1.7 we get
$H^{0}\left(\mathbb{S}_{N}^{+},I_{Y}\otimes\mathcal{U}(1)\right)\subset\operatorname{ker}(\Phi_{q})$
and in particular $\mathcal{Z}(\operatorname{ker}(\Phi_{q}))\subset Y_{supp}$,
where $\mathcal{Z}(\operatorname{ker}(\Phi_{q}))$ is the common zero locus of
global sections in $\operatorname{ker}(\Phi_{q})$. If such common zero locus
is given by only one pure spinor $[p]\in\mathbb{S}_{N}^{+}$, then for any
$0$-dimensional subscheme $Y\subset X$ of length $2$ such that $[q]\in\langle
Y\rangle$ it holds $Y_{supp}=\\{[p]\\}$, that is $[q]$ lies on only one
tangent space, namely $T_{[p]}\mathbb{S}_{N}^{+}$, and $[q]$ is tangential-
identifiable.

######  Theorem 5.2.

For any $N\geq 6$ and $l\geq 3$, the tangent orbit
$\Theta_{l,N}\subset\tau(\mathbb{S}_{N}^{+})$ is tangential-identifiable.

###### Proof.

Fix $l=3:\frac{N}{2}$. From the above argument it is enough to prove that
$\mathcal{Z}\left(\operatorname{ker}(\Phi_{q_{l}})\right)=\\{[\mathbbm{1}]\\}$
for $[q_{l}]\in
T_{[\mathbbm{1}]}\mathbb{S}_{N}^{+}\setminus\mathbb{S}_{n+1}^{+}$ being the
representative of $\Theta_{l,N}$ as in (3.7):

$q_{l}=\sum_{i=1}^{l}e_{2i-1}\wedge e_{2i}\ \in\bigwedge^{2}E\ .$

From the Clifford apolarity, for any $f\in\bigwedge^{od}E^{\vee}$ we have
$\Phi_{q_{l}}(f)=C_{q_{l}}(f)_{|_{E}}+C_{f}(q_{l})_{|_{E^{\vee}}}$. First,
notice that for any $f=\sum_{r=1}^{N}\beta_{r}f_{r}\in E^{\vee}$ one gets

$C_{q_{l}}(f)=\sum_{r=1}^{N}\sum_{i=1}^{l}\beta_{r}\big{[}f_{r}\cdot(e_{2i-1}\wedge
e_{2i})\big{]}=\sum_{i=1}^{l}\big{(}\beta_{2i-1}e_{2i}-\beta_{2i}e_{2i-1}\big{)}\
,$

implying that $\operatorname{ker}(\Phi_{q_{l}})\cap E^{\vee}=\\{0\\}$.
Moreover, since $q_{l}\in\bigwedge^{2}E$, it is straightforward that

$\bigwedge^{od\geq 5}E^{\vee}:=\bigoplus_{k\geq
2}\bigwedge^{2k+1}E^{\vee}\subset\operatorname{ker}(\Phi_{q_{l}})\ .$

As the images $\Phi_{q_{l}}(\bigwedge^{2k+1}E^{\vee})$ are linearly
independent one to each other, we get

$\operatorname{ker}(\Phi_{q_{l}})=\bigwedge^{od\geq
5}E^{\vee}\oplus\left(\operatorname{ker}(\Phi_{q_{l}})\cap\bigwedge^{3}E^{\vee}\right)\
,$

thus $\mathcal{Z}\left(\operatorname{ker}(\Phi_{q_{l}})\right)$ is the
intersection of the common zero loci of the two summands above.
Consider $[a]\in\mathcal{Z}\big{(}\operatorname{ker}(\Phi_{q_{l}})\big{)}$
such that
$a=\sum_{s=0}^{\frac{N}{2}}\sum_{J\in\binom{[N]}{2s}}\alpha_{J}\mathbb{e}_{J}$.
From Lemma 2.6 it holds

$\langle V\cdot f,a\rangle=0\ \ \ \ \ ,\forall f\in\bigwedge^{od\geq
5}E^{\vee}\oplus\left(\operatorname{ker}(\Phi_{q_{l}})\cap\bigwedge^{3}E^{\vee}\right)\
.$

The same computation in (4.1) shows that for any index subset $I\subset[N]$
such that $|I|=2k+1\geq 5$ and for any basis vector $e_{\lambda}\in E$ it
holds $0=\langle
e_{\lambda}\cdot\mathbb{f}_{I},a\rangle=(-1)^{pos(\lambda,I)+1}\alpha_{I\setminus\\{\lambda\\}}$
where again $pos(\lambda,I)$ denotes the position of $\lambda$ in the ordered
index subset $I$. We deduce that in $a$ there is no summand indexed by a
subset $J$ such that $J\subset I$ and $|J|=|I|-1$ for any $|I|\geq 5$, hence
$a\in\mathbb{C}\oplus\bigwedge^{2}E\oplus\bigwedge^{n+1}E$. On the other hand,
the same computation as in (4.3) shows that for any $I\subset[N]$ such that
$|I|=3$ and any basis vector $f_{\nu}\in E^{\vee}$ such that $\nu\notin I$, it
holds $0=\langle
f_{\nu}\cdot\mathbb{f}_{I},a\rangle=(-1)^{pos(\nu,I\cup\\{\nu\\})+1}\alpha_{I\cup\\{\nu\\}}$,
hence $a$ has no summand in $\bigwedge^{N}E$ either. It follows

$\mathcal{Z}\left(\operatorname{ker}(\Phi_{q_{l}})\right)=\mathbb{P}\left(\mathbb{C}\oplus\bigwedge^{2}E\right)\cap\mathcal{Z}\left(\bigwedge^{3}E^{\vee}\cap\operatorname{ker}\big{(}\Phi_{q_{l}}\big{)}\right)\
.$

Clearly, $[\mathbbm{1}]\in\mathcal{Z}(\operatorname{ker}(\Phi_{q_{l}}))$ since
for any $\lambda\in\mathbb{C}$ and any $f\in\bigwedge^{3}E^{\vee}$ one has
$\langle f,v\cdot\lambda\rangle=\lambda\langle f,v\rangle=0$. Moreover, for
any $b\in\bigwedge^{2}E$ it holds that
$b\in\mathcal{Z}\left(\operatorname{ker}(\Phi_{q_{l}})\right)$ if and only if
$\lambda+b\in\mathcal{Z}\left(\operatorname{ker}(\Phi_{q_{l}})\right)$ for any
$\lambda\in\mathbb{C}$: in particular, it is enough to prove that
$\mathcal{Z}\left(\operatorname{ker}(\Phi_{q_{l}})\right)\cap\mathbb{P}(\bigwedge^{2}E)=\emptyset$
in order to conclude.
Let $[a]\in\mathcal{Z}\left(\operatorname{ker}(\Phi_{q_{l}})\right)$ be such
that $a=\sum_{\\{s\lneq t\\}\subset[N]}\alpha_{st}e_{s}\wedge e_{t}$. First,
consider an index subset $I\subset[N]$ such that $|I|=3$ and
$\\{2k-1,2k\\}\nsubseteq I$ for any $k\in[l]$: the condition
$C_{\mathbb{f}_{I}}(q_{l})_{|_{E^{\vee}}}=0$ implies
$\mathbb{f}_{I}\in\operatorname{ker}(\Phi_{q_{l}})$. Then, for any
$e_{\lambda}\in E$ one gets

$0=\langle
e_{\lambda}\cdot\mathbb{f}_{I},a\rangle=\delta_{\lambda,i_{1}}\alpha_{i_{2},i_{3}}-\delta_{\lambda,i_{2}}\alpha_{i_{1},i_{3}}+\delta_{\lambda,i_{3}}\alpha_{i_{1},i_{2}}\
,$

for $\delta_{xy}$ being the Kronecker symbol. Since $N\geq 6$, given any two
distinct indices $\\{i\lneq j\\}\subset[N]$ such that
$\\{i,j\\}\neq\\{2k-1,2k\\}$ for any $k\in[l]$, one can always find a third
index $r\in[N]\setminus\\{i,j\\}$ such that $\\{2k-1,2k\\}\nsubseteq
I=\\{i,j,r\\}$ for any $k\in[l]$. Thus for any $\\{i,j\\}\neq\\{2k-1,2k\\}$ it
holds $\alpha_{ij}=0$ and

$\mathcal{Z}\left(\operatorname{ker}(\Phi_{q_{l}})\right)\cap\mathbb{P}\left(\bigwedge^{2}E\right)\subset\left\\{[a]=\left[\sum_{k=1}^{l}\alpha_{2k-1,2k}e_{2k-1}\wedge
e_{2k}\right]\right\\}\ .$

Now, for any $\\{k\lneq h\\}\subset[l]$ and any
$r\in[N]\setminus\\{2k-1,2k,2h-1,2h\\}$ consider
$\mathbb{f}_{2k-1,2k,r}-\mathbb{f}_{2h-1,2h,r}\in\bigwedge^{3}E^{\vee}$: it is
a straightforward count that
$\mathbb{f}_{2k-1,2k,r}-\mathbb{f}_{2h-1,2h,r}\in\operatorname{ker}(\Phi_{q_{l}})$.
In particular, $0=\langle
e_{r}\cdot(\mathbb{f}_{2k-1,2k,r}-\mathbb{f}_{2h-1,2h,r}),e_{r}\cdot
a\rangle=\alpha_{2k-1,2k}-\alpha_{2h-1,2h}$, hence it holds

$\alpha_{2k-1,2k}=\alpha_{2h-1,2h}\ \ \ \ \ ,\forall\\{h\lneq k\\}\subset[l]\
.$

It follows that $a=\alpha\cdot q_{l}$ for some $\alpha\in\mathbb{C}$. But
$[q_{l}]\notin\mathbb{S}_{N}^{+}$ (since $l\geq 3$), thus it has to be
$\alpha=0$ and
$\mathcal{Z}\left(\operatorname{ker}(\Phi_{q_{l}})\right)\cap\mathbb{P}(\bigwedge^{2}E)=\emptyset$.
We conclude that
$\mathcal{Z}\left(\operatorname{ker}(\Phi_{q_{l}})\right)=\\{[\mathbbm{1}]\\}$.
∎

##  6 Dimensions of $\operatorname{Spin}_{2N}$–orbits in
$\sigma_{2}(\mathbb{S}_{N}^{+})$

In this section we compute the dimensions of each orbit in the secant variety
of lines to a Spinor variety. We have postponed this computation as we apply
tangential-identifiability for computing dimensions in the tangent branch. We
assume the setting in Remark 1.1.
From [Zak93, Theorem 1.4] the secant orbit $\Sigma_{\frac{N}{2},N}$ is dense
of dimension $\dim\Sigma_{\frac{N}{2},N}=\dim\sigma_{2}(\mathbb{S}_{N}^{+})$
as in (1.7). Moreover, for $N\geq 6$ the tangential variety has codimension
$1$, hence its dense orbit $\Theta_{\frac{N}{2},N}$ has dimension
$\dim\Theta_{\frac{N}{2},N}=\dim\tau(\mathbb{S}_{N}^{+})=N(N-1)$.
Recall that for $N\leq 5$ the Spinor variety $\mathbb{S}_{N}^{+}$ has diameter
$2$, and the distance–$2$ orbit $\Sigma_{2,N}$ either does not exist (for
$N\leq 3$) or it is the dense one. In this respect, in the following we assume
$N\geq 6$.

######  Lemma 6.1.

For any $N\geq 6$ and any $l=2:\frac{N}{2}-1$, the fibration

$\begin{matrix}\xi:&\Sigma_{l,N}&\longrightarrow&\operatorname{OGr}(N-2l,V)\\\
&[a+b]&\mapsto&H_{a+b}&\stackrel{{\scriptstyle\text{Cor.}\ \ref{cor:fiber of
Sigma_l}}}{{=}}H_{a}\cap H_{b}\end{matrix}$

has fibers isomorphic to the dense orbit on the smaller Spinor variety
$\mathbb{S}_{2l}^{+}$, namely
$\xi^{-1}(H)\simeq\Sigma_{l,2l}\subset\sigma_{2}(\mathbb{S}_{2l}^{+})$. In
particular,

$\overline{\xi^{-1}(H)}\simeq\sigma_{2}(\mathbb{S}_{2l}^{+})\ .$

###### Proof.

Consider the distance-$l$ orbit
$\Sigma_{l,N}=\operatorname{Spin}_{2N}\cdot[\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}]$
and the above fibration $\xi$. By homogeneity, it is enough to determine the
fiber at the subspace $\xi([\mathbb{e}_{[N]}+\mathbb{e}_{[N-2l]}])=E\cap
E_{N-2l}=\langle e_{1},\ldots,e_{N-2l}\rangle_{\mathbb{C}}$. We set
$V^{\prime}=\langle
e_{N-2l+1},\ldots,e_{N},f_{N-2l+1},\ldots,f_{N}\rangle_{\mathbb{C}}$ and
$E^{\prime}=V^{\prime}\cap E$.
Let $[a+b]\in\xi^{-1}(E\cap E_{N-2l})$: then $E\cap E_{l}\subset H_{a}$ (resp.
$E\cap E_{l}\subset H_{b}$) and, by definition of the maximal fully isotropic
subspaces as kernels of (1.3), we can write $[a]=[\mathbb{e}_{[N-2l]}\wedge
w_{a}]$ (resp. $[b]=[\mathbb{e}_{[N-2l]}\wedge w_{b}]$) for some
$w_{a}\in\bigwedge^{ev}E^{\prime}$ (resp. $w_{b}\in\bigwedge^{ev}E^{\prime}$).
In particular, since $w_{a}$ and $w_{b}$ are defined by $2l$ linearly
independent columns in the matrices describing $H_{a}$ and $H_{b}$, they
corresponds to maximal isotropic subspaces
$H_{a}^{\prime},H_{b}^{\prime}\subset\operatorname{OGr}^{+}(2l,V^{\prime})$,
hence $[w_{a}],[w_{b}]\in\mathbb{S}_{2l}^{+}$. Finally, the condition
$H_{a}\cap H_{b}=E\cap E_{N-2l}$ implies $H_{a}^{\prime}\cap
H_{b}^{\prime}=(0)$, hence $[w_{a}]$ and $[w_{b}]$ have maximum Hamming
distance $d([w_{a}],[w_{b}])=l$ in $\mathbb{S}_{2l}^{+}$. Therefore the
injective wedge-multiplication map $(\mathbb{e}_{[N-2l]}\wedge\bullet)\ :\
\Sigma_{l,2l}\rightarrow\Sigma_{l,N}$ in (4.5) gives the biregular isomorphism

$\displaystyle\xi^{-1}(E\cap E_{N-2l})$
$\displaystyle=\left\\{[a+b]\in\Sigma_{l,N}\ |\ H_{a}\cap H_{b}=E\cap
E_{N-2l}\right\\}$
$\displaystyle=\left\\{[\mathbb{e}_{[N-2l]}\wedge(w_{a}+w_{b})]\in\Sigma_{l,N}\
|\ [w_{a}],[w_{b}]\in\mathbb{S}_{2l}^{+},\ d([w_{a}],[w_{b}])=l\right\\}$
$\displaystyle\simeq\left\\{[w_{a}+w_{b}]\in\sigma_{2}^{\circ}(\mathbb{S}_{2l}^{+})\
|\ d([w_{a}],[w_{b}])=l\right\\}$ $\displaystyle=\Sigma_{l,2l}\ .$

∎

######  Proposition 6.2.

For any $N\geq 6$ and $l=2:\frac{N}{2}-1$, the secant orbit
$\Sigma_{l,N}\subset\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$ has dimension

$\dim\Sigma_{l,N}=\begin{cases}\frac{N(N-1)}{2}+4N-15&\text{if }l=2\\\
\frac{N(N-1)}{2}+l(2N-1)-2l^{2}+1&\text{if }l\geq 3\ .\end{cases}$

###### Proof.

Consider the fibration $\xi$ in Lemma 6.1. From the fiber dimension theorem we
get

$\dim\Sigma_{l,N}=\dim\operatorname{OGr}(N-2l,V)+\dim\sigma_{2}(\mathbb{S}_{2l}^{+})\
.$

In general, the orthogonal Grassmannian $\operatorname{OGr}(r,M)$ coincides
with the kernel of the global section $s_{q}\in
H^{0}(\operatorname{Gr}(r,M),\operatorname{Sym}^{2}\mathcal{U})$ induced by
the quadratic form $q$ on $\mathbb{C}^{M}$, where $\mathcal{U}$ is the
rank-$r$ universal bundle on $\operatorname{Gr}(r,M)$. Thus

$\dim\operatorname{OGr}(r,M)=\dim\operatorname{Gr}(r,M)-\operatorname{Rk}\left(\operatorname{Sym}^{2}(\mathcal{U})\right)=r(M-r)-\binom{r+1}{2}\
.$

The thesis follows by substituting $r=N-2l$ and $M=2N$, and recovering
$\dim\sigma_{2}(\mathbb{S}_{2l}^{+})$ from (1.7). ∎

The above computation puts on light a particular feature of the second-to-last
secant orbit. In the following result we need $N\geq 8$ in order to get an
intermediate proper secant orbit between $\Sigma_{2,N}$ and
$\Sigma_{\frac{N}{2},N}$.

######  Corollary 6.3.

For any $N\geq 8$, the closure of the second-to-last orbit
$\Sigma_{\frac{N-2}{2},N}$ is a divisor in $\sigma_{2}(\mathbb{S}_{N}^{+})$
parametrized by the vanishing of a pfaffian. Indeed, up to chart-changing, all
pure spinors $[a]\in\mathbb{S}_{N}^{+}$ such that
$d([\mathbb{e}_{[N]}],[a])=\frac{N-2}{2}$ correspond to maximal fully
isotropic subspaces $H_{a}$ described by matrices
${\tiny\begin{bmatrix}I_{N}\\\ A\end{bmatrix}}$ where
$\operatorname{Rk}(A)=N-\dim(E\cap H_{a})=N-2$.

###### Proof.

The thesis “being a divisor” is a straightforward count from Proposition 6.2.
We show that the closure of $\Sigma_{\frac{N-2}{2},N}$ is parametrized by the
vanishing of a pfaffian.
The pure spinors $[a]\in\mathbb{S}_{N}^{+}$ having Hamming distance $l$ from
$[\mathbb{e}_{[N]}]$ correspond to subspaces
$H_{a}\in\operatorname{OGr}^{+}(N,V)$ such that $\dim(E\cap H_{a})=N-2l$. Up
to chart-changing, we may assume that $H_{a}$ is described by the matrix
$\scriptsize{\begin{bmatrix}I_{N}\\\ A\end{bmatrix}}$ for a certain
$A\in\bigwedge^{2}\mathbb{C}^{N}$: in particular,
$\operatorname{Rk}(A)=N-\dim(E\cap H_{a})=2l$. It follows that the pure
spinors having Hamming distance $l=\frac{N-2}{2}$ from $[\mathbb{e}_{[N]}]$
are described (up to chart-changing) by skew–symmetric matrices of rank $N-2$,
that is they are parametrized by the vanishing of the pfaffian of such
matrices. Now, given the pure spinor $[\mathbb{e}_{[N]}]$, the subvarieties

$\mathcal{F}_{\mathbb{e}_{[N]},\frac{N-2}{2}}=\left\\{[\mathbb{e}_{[N]}+a]\
\big{|}\ [a]\in\mathbb{S}_{N}^{+},\
d([\mathbb{e}_{[N]}],[a])\leq\frac{N-2}{2}\right\\}\subset\overline{\Sigma_{\frac{N-2}{2},N}}\
,$ $\mathcal{F}_{\mathbb{e}_{[N]}}=\left\\{[\mathbb{e}_{[N]}+a]\ |\
[a]\in\mathbb{S}_{N}^{+}\right\\}\subset\sigma_{2}^{\circ}(\mathbb{S}_{N}^{+})$

are such that
$\mathcal{F}_{\mathbb{e}_{[N]},\frac{N-2}{2}}=\mathcal{F}_{\mathbb{e}_{[N]}}\cap
V(\operatorname{Pf}(A))$: more in general, for any pure spinor
$[b]\in\mathbb{S}_{N}^{+}$ it holds
$\mathcal{F}_{b,\frac{N-2}{2}}=\mathcal{F}_{b}\cap V(\operatorname{Pf}(A))$,
giving the rational isomorphism in the thesis. ∎

We are left with computing dimensions in the tangent branch for $N\geq 6$.
From Proposition 6.2 we know that $\Theta_{2,N}=\Sigma_{2,N}$ has dimension
$\frac{N(N-1)}{2}+4N-15$. Thus in the following we assume $N\geq 6$ and
$l=3:\frac{N-1}{2}-1$.

######  Remark.

In the following proof we use that the tangent orbits $\Theta_{l,N}$ for
$l\geq 3$ are tangentially-identifiable (cf. Theorem 5.2): each point
$[q]\in\Theta_{l,N}$ lies on a unique tangent space
$T_{[a]}\mathbb{S}_{N}^{+}$.

######  Proposition 6.4.

For any $N\geq 6$ and any $l=3:\frac{N}{2}-1$, the tangent orbit
$\Theta_{l,N}\subset\tau(\mathbb{S}_{N}^{+})$ has dimension

$\displaystyle\dim\Theta_{l,N}=\frac{N(N-1)}{2}+l(2N-1)-2l^{2}\ .$

In particular, for such $N,l$, the closure $\overline{\Theta_{l,N}}$ is a
divisor in $\overline{\Sigma_{l,N}}$.

###### Proof.

Given $[\bigwedge^{2}\mathbb{C}^{N}]_{2l}$ the space of rank–$2l$
skew–symmetric matrices of size $N$, from Sec. 3.2 we know that for any
$l=2:\frac{N}{2}$ it holds

$\Theta_{l,N}=\bigcup_{[a]\in\mathbb{S}_{N}^{+}}\left[T_{[a]}\mathbb{S}_{N}^{+}\right]_{2l}\simeq\bigcup_{[a]\in\mathbb{S}_{N}^{+}}\left[\bigwedge^{2}\mathbb{C}^{N}\right]_{2l}\
.$

For $l\geq 3$, any tangent $[q]\in\Theta_{l,N}$ belongs to a unique tangent
space $T_{[a]}\mathbb{S}_{N}^{+}$, hence

$\dim\Theta_{l,N}=\dim\mathbb{S}_{N}^{+}+\dim\left[\bigwedge^{2}\mathbb{C}^{N}\right]_{2l}\
\ \ \ \ ,\ \forall l\geq 3\ .$

The space $[\bigwedge^{2}\mathbb{C}^{N}]_{2l}$ is the
$\operatorname{GL}(N)$-orbit in $\mathbb{C}^{N}\otimes\mathbb{C}^{N}$ of the
skew–symmetric matrix $C_{l}$ in (3.8) having stabilizer isomorphic to
$\operatorname{Sp}(2l)\times\operatorname{GL}(N-2l)\times(\mathbb{C}^{2l}\otimes\mathbb{C}^{N-2l})$.
Therefore one gets
$\dim\left[\bigwedge^{2}\mathbb{C}^{N}\right]_{2l}=l(2N-1)-2l^{2}$ and the
thesis straightforwardly follows. ∎

######  Example 6.5.

The Spinor variety
$\mathbb{S}_{6}^{+}\subset\mathbb{P}(\bigwedge^{ev}\mathbb{C}^{6})\simeq\mathbb{P}^{31}$
has diameter $3$. Set $G=\operatorname{Spin}(12)$ and
$v_{\omega_{6}}=e_{1}\wedge\ldots\wedge e_{6}$. The secant variety
$\sigma_{2}(\mathbb{S}_{6}^{+})$ stratifies in the
$\operatorname{Spin}(12)$–orbit closures

$\mathbb{S}_{6}^{+}\subset\overline{\Sigma_{2,6}}\subset\overline{\Theta_{3,6}}\subset\overline{\Sigma_{3,6}}=\sigma_{2}(\mathbb{S}_{6}^{+})$

of dimensions respectively $15,24,30,31$. This poset and the dimensions were
already known to J.M. Landsberg and L. Manivel [LM01] in the context of
Legendrian varieties. Actually, as confirmed by the authors (which we thank
for the confrontation), our arguments allow to recognize a misprint in [LM01,
Proposition 5.10], where the dimension of the orbit $\sigma_{+}$ (in the
authors’ notation, corresponding to our $\Sigma_{2,6}$ for $m=4$) is $5m+4$
instead of $5m+3$.

######  Example 6.6.

The Spinor variety
$\mathbb{S}_{8}^{+}\subset\mathbb{P}(\bigwedge^{ev}\mathbb{C}^{8})\simeq\mathbb{P}^{127}$
has diameter $4$: this is the only case in which the inclusion
$\overline{\Theta_{3,8}}\subset\overline{\Theta_{4,8}}\cap\overline{\Sigma_{3,8}}$
actually is an equality.

$\mathbb{S}_{8}^{+}$$28$$\Theta_{2,8}=\Sigma_{2,8}$$45$$\Theta_{3,8}$$55$$\Sigma_{3,8}$$56$$\Theta_{4,8}$$\Sigma_{4,8}$$57$
Figure 2: Poset graph of the $\operatorname{Spin}_{16}$–orbits in
$\sigma_{2}(\mathbb{S}_{8}^{+})$, and their dimensions.

##  7 The $2$–nd Terracini locus of $\mathbb{S}_{N}^{+}$

In the following we determine the Terracini locus, introduced in [BC21,
BBS20]. Its importance relies in the fact that it gives information on the
singularities of points lying on bisecant lines.
The $2$–nd Terracini locus of $\mathbb{S}_{N}^{+}$ is

$\operatorname{Terr}_{2}(\mathbb{S}_{N}^{+}):=\overline{\left\\{\\{p_{1},p_{2}\\}\in
Hilb_{2}(\mathbb{S}_{N}^{+})\ |\ \dim\langle
T_{p_{1}}\mathbb{S}_{N}^{+},T_{p_{2}}\mathbb{S}_{N}^{+}\rangle\lneq\dim\sigma_{2}(\mathbb{S}_{N}^{+})\right\\}}\
.$

We also recall the definition of the $2$–nd abstract secant variety of
$\mathbb{S}_{N}^{+}$

$Ab\sigma_{2}(\mathbb{S}_{N}^{+}):=\left\\{(\mathcal{Z},p)\in
Hilb_{2}(\mathbb{S}_{N}^{+})\times\mathbb{P}\left(\bigwedge^{ed}E\right)\ |\
p\in\langle\mathcal{Z}\rangle\right\\}\ ,$

together with the natural projections onto the two factors
$\pi_{1}:Ab\sigma_{2}(\mathbb{S}_{N}^{+})\rightarrow
Hilb_{2}(\mathbb{S}_{N}^{+})$ and
$\pi_{2}:Ab\sigma_{2}(\mathbb{S}_{N}^{+})\rightarrow\sigma_{2}(\mathbb{S}_{N}^{+})\subset\mathbb{P}(\bigwedge^{ev}E)$.
Notice that, with this definition, the $2$–nd abstract secant variety is
smooth.
The $2$–nd Terracini locus tells us where the differential of the projection
from the abstract secant variety onto the secant variety drops rank: this
information, combined with the identifiability, allows to deduce the
smoothness on the locus of points which are both identifiable and outside the
Terracini locus. In the following we consider $N\geq 7$ as for $N\leq 6$ the
secant variety of lines coincides with the ambient space.

######  Theorem 7.1.

For any $N\geq 7$, the second Terracini locus
$\operatorname{Terr}_{2}(\mathbb{S}_{N}^{+})$ of the Spinor variety
$\mathbb{S}_{N}^{+}$ corresponds to the distance–$2$ orbit closure
$\overline{\Sigma_{2,N}}$. More precisely, in the above notation, it holds

$\operatorname{Terr}_{2}(\mathbb{S}_{N}^{+})=\left(\pi_{1}\circ\pi_{2}^{-1}\right)\left(\overline{\Sigma_{2,N}}\right)\
.$

###### Proof.

Given a point $[a+b]\in\sigma_{2}(\mathbb{S}_{N}^{+})$ for certain
$[a],[b]\in\mathbb{S}_{N}^{+}$, we show that $\dim\langle
T_{[a]}\mathbb{S}_{N}^{+},T_{[b]}\mathbb{S}_{N}^{+}\rangle$ drops if and only
if $[a+b]\in\overline{\Sigma_{2,N}}=\mathbb{S}_{N}^{+}\sqcup\Sigma_{2,N}$,
that is if and only if $d([a],[b])\leq 2$.
If $d([a],[b])=1$, then any point on the line
$L([a],[b])\subset\mathbb{S}_{N}^{+}$ is direction of the curve defined by
such line, that is $L([a],[b])\subset T_{[a]}\mathbb{S}_{N}^{+}\cap
T_{[b]}\mathbb{S}_{N}^{+}$.
If $d([a],[b])=2$, then there exists $[c]\in\mathbb{S}_{N}^{+}$ such that
$L([a],[c]),L([c],[b])\subset\mathbb{S}_{N}^{+}$. In particular, from the
previous case we deduce $[c]\in T_{[a]}\mathbb{S}_{N}^{+}\cap
T_{[b]}\mathbb{S}_{N}^{+}$.
On the other hand, if the dimension drops, then there exists a common non-zero
tangent point $[x]\in T_{[a]}\mathbb{S}_{N}^{+}\cap
T_{[b]}\mathbb{S}_{N}^{+}$. In particular, $[x]$ is not tangential-
identifiable, hence $[x]\in\Theta_{2,N}$. From the definition (3.6) of
$\Theta_{2,N}\cap T_{[a]}\mathbb{S}_{N}^{+}$ as the set of rank–$4$ skew-
symmetric matrices in $T_{[a]}\mathbb{S}_{N}^{+}$, we know that $[x]=[y+z]$
for $[y],[z]\in\Theta_{1,N}\cap
T_{[a]}\mathbb{S}_{N}^{+}=\mathbb{S}_{N}^{+}\cap T_{[a]}\mathbb{S}_{N}^{+}$
skew-symmetric matrices of rank $2$. In particular, $[a+y]$ and $[a+z]$ lie in
$\mathbb{S}_{N}^{+}$, that is $d([a],[y])=d([a],[z])=1$. The same argument
shows that also $d([b],[y])=d([b],[z])=1$. We conclude that $d([a],[b])\leq
2$, hence the thesis.

∎

##  8 Results on the singular locus of $\sigma_{2}(\mathbb{S}_{N}^{+})$

This section is devoted to study the singular locus of the secant variety of
lines $\sigma_{2}(\mathbb{S}_{N}^{+})$ to a Spinor variety
$\mathbb{S}_{N}^{+}$. We use results from previous sections.

######  Remark.

As defective cases, the secant varieties
$\sigma_{2}(\mathbb{S}_{4}^{+})=\tau(\mathbb{S}_{4}^{+})=\mathbb{P}^{7}$ and
$\sigma_{2}(\mathbb{S}_{5}^{+})=\tau(\mathbb{S}_{5}^{+})=\mathbb{P}^{15}$
overfill the ambient space. On the other hand, the secant variety
$\sigma_{2}(\mathbb{S}_{6}^{+})=\mathbb{P}^{31}$ perfectly fills the ambient
space (hence it is smooth), but the tangential variety
$\tau(\mathbb{S}_{6}^{+})$ is a quartic hypersurface in it. Accordingly to
Remark 5.1, $\mathbb{S}_{6}^{+}$ has been widely studied in [LM01, LM07] and
it has been proven that
$\operatorname{Sing}(\tau(\mathbb{S}_{6}^{+}))=\overline{\Sigma_{2,6}}$
(corresponding to $\sigma_{+}$ in [LM07]).

According to the above remark, we assume $N\geq 7$, so that
$\sigma_{2}(\mathbb{S}_{N}^{+})\subsetneq\mathbb{P}^{2^{N-1}-1}$. In the
following we consider the alternative definition of abstract secant variety

$Ab\sigma_{2}(\mathbb{S}_{N}^{+}):=\overline{\left\\{([a],[b],[q])\in\big{(}\mathbb{S}_{N}^{+}\big{)}^{2}_{/\mathfrak{S}_{2}}\times\mathbb{P}\left(\bigwedge^{ev}E\right)\
\bigg{|}\ [q]\in\langle[a],[b]\rangle\right\\}}\ ,$

which is smooth only outside the preimage of the diagonal
$\Delta_{X}\subset\big{(}\mathbb{S}_{N}^{+}\big{)}^{2}_{/\mathfrak{S}_{2}}$
via the projection onto the first factor. In particular, given $\pi$ the
projection onto the second factor, the preimage
$\pi^{-1}(\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\overline{\Sigma_{2,N}})$ is
smooth in $Ab\sigma_{2}(\mathbb{S}_{N}^{+})$.

######  Lemma 8.1.

For any $N\geq 7$, the distance–$2$ orbit $\Sigma_{2,N}$ lies in the singular
locus $\operatorname{Sing}(\sigma_{2}(\mathbb{S}_{N}^{+}))$ of the secant
variety of lines to the Spinor variety $\mathbb{S}_{N}^{+}$. In particular,

$\overline{\Sigma_{2,N}}\subset\operatorname{Sing}\left(\sigma_{2}(\mathbb{S}_{N}^{+})\right)\
.$

###### Proof.

Clearly, the Spinor variety is singular in its secant variety of lines, hence
we just have to prove the singularity of the distance–$2$ orbit
$\Sigma_{2,N}$. We assume by contradiction that $\Sigma_{2,N}$ is smooth: then
the poset in Theorem 3.12 implies that all the open subset
$\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\mathbb{S}_{N}^{+}$ is smooth.
Consider the projection from the abstract secant variety onto the second
factor

$\begin{matrix}\pi:&Ab\sigma_{2}(\mathbb{S}_{N}^{+})&\longrightarrow&\sigma_{2}(\mathbb{S}_{N}^{+})\\\
&([a],[b],[q])&\mapsto&[q]\end{matrix}\ .$

The (tangential-)identifiability of the orbits $\Sigma_{l,N}$ and
$\Theta_{l,N}$ for $l\geq 3$ (cf. Theorem 4.5 and Theorem 5.2) implies that
the restriction of $\pi$ to the open subset
$\pi^{-1}(\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\overline{\Sigma_{2,N}})$ is
a bijection of smooth open subsets, hence it is an isomorphism. On the other
hand, the orbit $\Sigma_{2,N}$ is unidentifiable and any of its points has
$6$–dimensional decomposition locus (cf. Proposition 4.3), thus the
differential $d(\pi)_{([a],[b],[q])}$ drops rank exactly at the points in the
preimage $\pi^{-1}(\overline{\Sigma_{2,N}})$.
It follows that the restriction

${\pi}_{|}:Ab\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\pi^{-1}(\mathbb{S}_{N}^{+})\longrightarrow\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\mathbb{S}_{N}^{+}$

is a morphism of smooth varieties of the same dimension: in particular, the
locus of points where the rank of the differential drops is a (determinantal)
divisor. We show that the preimage

$\left\\{([a],[b],[q])\in
Ab\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\pi^{-1}(\mathbb{S}_{N}^{+})\ |\
\operatorname{Rk}d(\pi)_{([a],[b],[q])}<N(N-1)\right\\}={\pi_{|}}^{-1}\left(\overline{\Sigma_{2,N}}\right)$

cannot be a divisor (leading to a contradiction). Indeed, from Proposition 6.2
we know that $\dim\Sigma_{2,N}=\frac{N(N-1)}{2}+4N-15$, and from Proposition
4.3 that the decomposition locus of any point in $\Sigma_{2,N}$ is
$6$–dimensional, thus from the fiber dimension theorem we get
$\dim\pi_{|}^{-1}(\overline{\Sigma_{2,N}})=\dim\Sigma_{2,N}+6=\frac{N(N-1)}{2}+4N-9$
and

$\dim
Ab\sigma_{2}(\mathbb{S}_{N}^{+})-\dim\pi_{|}^{-1}(\overline{\Sigma_{2,N}})=\frac{N(N-1)}{2}-4N+10\stackrel{{\scriptstyle
N\geq 7}}{{\gneq}}1\ .$

Thus ${\pi}_{|}^{-1}(\overline{\Sigma_{2,N}})$ cannot be a divisor for $N\geq
7$, giving the contradiction. We deduce the inclusion
$\Sigma_{2,N}\subset\operatorname{Sing}(\sigma_{2}(\mathbb{S}_{N}^{+}))$. ∎

######  Lemma 8.2.

For any $N\geq 7$ and any $l=3:\frac{N}{2}$, the secant orbit $\Sigma_{l,N}$
is smooth in the secant variety of lines $\sigma_{2}(\mathbb{S}_{N}^{+})$. In
particular, the singular locus of the secant variety of lines to the Spinor
variety $\mathbb{S}_{N}^{+}$ lies in the tangential variety:

$\operatorname{Sing}\left(\sigma_{2}(\mathbb{S}_{N}^{+})\right)\subset\tau(\mathbb{S}_{N}^{+})\
.$

###### Proof.

From Proposition 4.5 we know that the secant orbits $\Sigma_{l,N}$ for $l=3:N$
are identifiable. Moreover, from the poset in Figure 1 it holds

$\Sigma_{3,N}\sqcup\ldots\sqcup\Sigma_{\frac{N}{2},N}=\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\tau(\mathbb{S}_{N}^{+})\
.$

Then the projection
$\pi:Ab\sigma_{2}(\mathbb{S}_{N}^{+})\rightarrow\sigma_{2}(\mathbb{S}_{N}^{+})$
from the abstract secant variety onto the second factor restricts to a
bijection

$\pi^{\prime}:\pi^{-1}\bigg{(}\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\tau(\mathbb{S}_{N}^{+})\bigg{)}\rightarrow\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\tau(\mathbb{S}_{N}^{+})\
.$

Finally, the points in
$\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\tau(\mathbb{S}_{N}^{+})$ are outside
the Terracini locus $\operatorname{Terr}_{2}(\mathbb{S}_{N}^{+})$ (cf. Theorem
7.1), thus for any point
$[p+q]\in\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\tau(\mathbb{S}_{N}^{+})$ it
holds $T_{p}\mathbb{S}_{N}^{+}\cap T_{q}\mathbb{S}_{N}^{+}=\\{0\\}$. In
particular, for such points the differential $d\pi^{\prime}_{([p],[q],[p+q])}$
maps $T_{p}\mathbb{S}_{N}^{+}\times T_{q}\mathbb{S}_{N}^{+}$ to
$T_{p}\mathbb{S}_{N}^{+}\oplus T_{q}\mathbb{S}_{N}^{+}$, hence it is
injective. It follows that $\pi^{\prime}$ is an isomorphism and the open
subset $\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\tau(\mathbb{S}_{N}^{+})$ is
smooth in the secant variety of lines. ∎

Collecting Lemma 8.1 and Lemma 8.2 we get the following bound on the singular
locus.

######  Corollary 8.3.

For any $N\geq 7$, the singular locus of the secant variety of lines
$\sigma_{2}(\mathbb{S}_{N}^{+})$ to the Spinor variety $\mathbb{S}_{N}^{+}$ is
such that

$\overline{\Sigma_{2,N}}\subset\operatorname{Sing}\left(\sigma_{2}(\mathbb{S}_{N}^{+})\right)\subset\tau(\mathbb{S}_{N}^{+})\
.$

We conclude this work with a conjecture on what the singular locus of
$\sigma_{2}(\mathbb{S}_{N}^{+})$ actually is, motivated by the arguments
apprearing in the appendix.

######  Conjecture 8.4.

For any $N\geq 7$, the singular locus of the secant variety of lines to the
Spinor variety $\mathbb{S}_{N}^{+}$ is the closure of the distance–$2$ orbit,
i.e.

$\operatorname{Sing}\left(\sigma_{2}(\mathbb{S}_{N}^{+})\right)=\overline{\Sigma_{2,N}}=\mathbb{S}_{N}^{+}\sqcup\Sigma_{2,N}\
.$

## Appendix

The arguments in this section are inspired from [Ver01, Ver09, Ull16]: the
author thanks L. Manivel for suggesting these references. They have been
investigated in the attempt of proving the smoothness of both secant and
tangent orbits of distance greater or equal than $3$ in the secant variety of
lines to a Spinor variety. The idea was to prove that the secant bundle gives
a desingularization of the secant variety, but we have just been able to
exhibit a bijection (as sets) between dense subsets.
We recall that a line bundle $\mathcal{L}$ on a smooth projective variety $X$
is $k$–very ample if every $0$–dimensional subscheme $\mathcal{Z}\subset X$ of
length $k+1$ imposes independent conditions on $\mathcal{L}$, or equivalently
if the restriction map $H^{0}(X,\mathcal{L})\rightarrow
H^{0}(X,\mathcal{L}\otimes\mathcal{O}_{\mathcal{Z}})$ is surjective: in
particular, $\mathcal{L}$ is $1$–very ample if and only if it is very ample,
and $\mathcal{L}$ is $k$–very ample if and only if there does not exist a
$0$–dimensional subscheme $\mathcal{Z}\subset X$ of length $k+1$ lying on a
linear subspace
$\mathbb{P}^{k-1}\subset\mathbb{P}(H^{0}(X,\mathcal{L})^{\vee})$.

######  Remark 8.5.

The line bundle $\mathcal{O}_{\mathbb{S}_{N}^{+}}(1)$ on the Spinor variety
$\mathbb{S}_{N}^{+}$ is very ample, giving the embedding in
$\mathbb{P}\left(H^{0}(\mathbb{S}_{N}^{+},\mathcal{O}_{\mathbb{S}_{N}^{+}}(1))^{\vee}\right)\simeq\mathbb{P}\left(\bigwedge^{ev}E\right)$,
but not $2$–very ample. Indeed, it would be $2$–very ample if and only if
there would not exist a $0$–dimensional subscheme of $\mathbb{S}_{N}^{+}$ of
length $3$ lying on a line $\mathbb{P}^{1}\subset\mathbb{P}(\bigwedge^{ev}E)$.
But we know that the Spinor variety contains lines, defined by pairs of pure
spinors having Hamming distance $1$.

#### Secant bundle.

Let $\mathcal{L}=\mathcal{O}(1)$ be the very ample line bundle on
$X:=\mathbb{S}_{N}^{+}$, and let $Hilb_{2}(X)$ be the Hilbert scheme of
$0$-dimensional subschemes of $\mathbb{S}_{N}^{+}$ of length $2$. We denote by
$\mathcal{E}_{\mathcal{L}}$ the locally free sheaf of rank $2$ on
$Hilb_{2}(X)$ having fibers
$\left(\mathcal{E}_{\mathcal{L}}\right)_{\mathcal{Z}}=H^{0}(X,\mathcal{L}\otimes\mathcal{O}_{\mathcal{Z}})$:
formally,
$\mathcal{E}_{\mathcal{L}}:=(\pi_{2})_{\ast}(\pi_{1}^{\ast})(\mathcal{L})$
where $\pi_{1}$ and $\pi_{2}$ are the natural projections from the universal
family $\Phi:=\\{(x,\mathcal{Z})\in X\times Hilb_{2}(X)\ |\
x\in\mathcal{Z}\\}$ onto $X$ and $Hilb_{2}(X)$ respectively. From the
projection formula, one gets the following global sections of sheaves:

$\displaystyle H^{0}(Hilb_{2}(X),\mathcal{E}_{\mathcal{L}})$
$\displaystyle=H^{0}(\Phi,(\pi_{X})^{\ast}\mathcal{L})=H^{0}(X,\mathcal{L}\otimes(\pi_{X})_{\ast}\mathcal{O}_{\Phi})=H^{0}(X,\mathcal{L})\
,$

where the last equality follows since $\pi_{X}$ is a proper projection and
$(\pi_{X})_{\ast}\mathcal{O}_{\Phi}=\mathcal{O}_{X}$.
By pushing forward via $\pi_{H}$ the morphism
$H^{0}(X,\mathcal{L})\otimes\mathcal{O}_{\Phi}\rightarrow(\pi_{X})^{\ast}\mathcal{L}$
(of sheaves on $\Phi$) one gets the evaluation morphism of sheaves on
$Hilb_{2}(X)$

$ev:H^{0}(X,\mathcal{L})\otimes\mathcal{O}_{Hilb_{2}(X)}\longrightarrow\mathcal{E}_{\mathcal{L}}\
,$

defined on the fibre as the restriction: for any $\mathcal{Z}\in Hilb_{2}(X)$
it holds

$\begin{matrix}ev_{\mathcal{Z}}:&H^{0}(X,\mathcal{L})&\longrightarrow&\left(\mathcal{E}_{\mathcal{L}}\right)_{\mathcal{Z}}&=H^{0}(X,\mathcal{L}\otimes\mathcal{O}_{\mathcal{Z}})\\\
&s&\mapsto&s_{|_{\mathcal{Z}}}\end{matrix}\ .$

As $\mathcal{L}$ is very ample, the evaluation

$ev_{\mathcal{Z}}:H^{0}(X,\mathcal{L})\twoheadrightarrow
H^{0}(X,\mathcal{\otimes}\mathcal{O}_{\mathcal{Z}})$

is a surjection for any $\mathcal{Z}\in Hilb_{2}(X)$, hence the morphism of
sheaves
$H^{0}(X,\mathcal{L})\otimes\mathcal{O}_{Hilb_{2}(X)}\rightarrow\mathcal{E}_{\mathcal{L}}$
is surjective. Moreover, again by very ampleness of $\mathcal{L}$, one can get
the injective morphism

$\begin{matrix}\psi:&Hilb_{2}(X)&\hookrightarrow&\operatorname{Gr}\big{(}2,H^{0}(X,\mathcal{L})\big{)}\\\
&\mathcal{Z}&\mapsto&\left[H^{0}(X,\mathcal{L})\twoheadrightarrow
H^{0}(X,\mathcal{L}\otimes\mathcal{O}_{\mathcal{Z}})\right]\end{matrix}\ .$

Now, we may think at $\mathbb{P}^{M}=\mathbb{P}(H^{0}(X,\mathcal{L})^{\vee})$
as to the $1$-dimensional quotients of $H^{0}(X,\mathcal{L})$, i.e.

$\mathbb{P}^{M}=\left\\{H^{0}(X,\mathcal{L})\twoheadrightarrow Q\ |\ \dim
Q=1\right\\}\ .$

We denote by $Q^{1}$ a $1$-dimensional vector space. Then one defines the
(first) secant bundle $\mathbb{P}\mathcal{E}_{\mathcal{L}}$ as the
$\mathbb{P}^{1}$-bundle on $Hilb_{2}(X)$ with fibers
$(\mathbb{P}\mathcal{E}_{\mathcal{L}})_{\mathcal{Z}}=\left\\{\left(\mathcal{Z},\left[H^{0}(X,\mathcal{L}\otimes\mathcal{O}_{\mathcal{Z}})\twoheadrightarrow\mathcal{Q}^{1}\right]\right)\right\\}$.
The following morphism is well-defined:

$\begin{matrix}f:&\mathbb{P}\mathcal{E}_{\mathcal{L}}&\longrightarrow&\mathbb{P}\left(H^{0}(X,\mathcal{L})^{\vee}\right)\\\
&\left(\mathcal{Z},\left[H^{0}(X,\mathcal{L}\otimes\mathcal{O}_{\mathcal{Z}})\twoheadrightarrow
Q^{1}\right]\right)&\mapsto&\left[H^{0}(X,\mathcal{L})\twoheadrightarrow
Q^{1}\right]\end{matrix}\ .$

Notice that $\left[H^{0}(X,\mathcal{L})\twoheadrightarrow
Q^{1}\right]\subset\operatorname{Im}(f)$ if and only if it factors through a
certain $H^{0}(X,\mathcal{L}\otimes\mathcal{O}_{\mathcal{Z}})$ for some
$\mathcal{Z}\in Hilb_{2}(X)$: in particular,

$f\left(\mathbb{P}\mathcal{E}_{\mathcal{L}}\right)=\sigma_{2}(X)\subset\mathbb{P}(H^{0}(X,\mathcal{L})^{\vee})\
.$

For any $\mathcal{Z}\in Hilb_{2}(X)$, the fibre
$\left(\mathbb{P}\mathcal{E}_{\mathcal{L}}\right)_{\mathcal{Z}}=\left\\{(\mathcal{Z},\left[H^{0}(X,\mathcal{L}\otimes\mathcal{O}_{\mathcal{Z}})\twoheadrightarrow
Q^{1}\right])\right\\}$ is bijectively mapped via $f$ to the secant line
$L(\mathcal{Z})\subset\sigma_{2}(X)$ intersecting $X$ in the points of
$\mathcal{Z}$: if $\mathcal{Z}=\\{p,q\\}$ is a reduced subscheme, then the
fibre at $\mathcal{Z}$ gives the bisecant line
$\left(\mathbb{P}\mathcal{E}_{\mathcal{L}}\right)_{\mathcal{Z}}\stackrel{{\scriptstyle
1:1}}{{\mapsto}}L(p,q)$, while if $\mathcal{Z}$ is a non-reduced subscheme
corresponding to $\\{p,v\\}$, where $p\in X$ and $v\in T_{p}X$, then the fibre
at $\mathcal{Z}$ gives the tangent line
$\left(\mathbb{P}\mathcal{E}_{\mathcal{L}}\right)_{\mathcal{Z}}\stackrel{{\scriptstyle
1:1}}{{\mapsto}}L_{v}(p)=p+tv$. Finally, in our setting, for any two distinct
subschemes $\mathcal{Z},\mathcal{Y}\in Hilb_{2}(X)$ the lines $L(\mathcal{Z})$
and $L(\mathcal{Y})$ coincide if and only if they lie in $X$, as the Spinor
variety is intersection of quadrics.

######  Lemma 8.6.

The restriction of $f$ to
$f^{-1}(\sigma_{2}(\mathbb{S}_{N}^{+})\setminus\overline{\Sigma_{2,N}})$ is a
bijection.

###### Proof.

By Theorem 4.5, the points in the secant orbits $\Sigma_{l,N}$ for $l\geq 3$
are identifiable, thus any two bisecant lines (given by points in
$\Sigma_{l,N}$ for $l\geq 3$, except for the two points in which they
intersect $\mathbb{S}_{N}^{+}$) do not intersect away from
$\mathbb{S}_{N}^{+}$. By Theorem 5.2, the points in the tangent orbits
$\Theta_{l,N}$ for $l\geq 3$ are tangential-identifiable, liying on a unique
tangent line, hence any two tangent lines do not intersect away from
$\overline{\Sigma_{2,N}}$. Finally, by the orbit partition of
$\sigma_{2}(\mathbb{S}_{N}^{+})$ in Theorem 3.12, since
$\Theta_{l,N}\neq\Sigma_{l,N}$ for any $l\geq 3$, a tangent line and a
bisecant line never intersect away from $\overline{\Sigma_{2,N}}$. It follows
that
$f:\mathbb{P}\mathcal{E}_{\mathcal{L}}\rightarrow\sigma(\mathbb{S}_{N}^{+})$
is a bijection away from $f^{-1}\left(\overline{\Sigma_{2,N}}\right)$. ∎

######  Remark.

We point out that Vermeire in [Ver09, Proposition 1.2, Theorem 2.2] and Ullery
in [Ull16, Lemma 1.1] prove the smoothness of $\sigma_{2}(X)\setminus X$ under
the assumption of the line bundle giving the embedding to be $3$-very ample:
this assumption is the one separating secant and tangent lines away from $X$.
In our case, the previous hypothesis is not satisfied (see Remark 8.5).
Neverthless, at a set-level, the identifibiability and the tangential-
identifiability allow to separate secant and tangent lines (away from
$\overline{\Sigma_{2,N}}$) as well. What is missing is that the above
restriction has injective differential, so that it would be an isomorphism.

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