# Spinor-Helicity Varieties

Yassine El Maazouz  Anaëlle Pfister and Bernd Sturmfels

###### Abstract

The spinor-helicity formalism in particle physics gives rise to natural
subvarieties in the product of two Grassmannians. These include two-step flag
varieties for subspaces of complementary dimension. Taking Hadamard products
leads to Mandelstam varieties. We study these varieties through the lens of
combinatorics and commutative algebra, and we explore their tropicalization,
positive geometry, and scattering correspondence.

##  1 Introduction

Given two matrices $\lambda$ and $\widetilde{\lambda}$ of format $k\times n$,
consider the property that the $k\times k$ matrix
$\lambda\cdot\widetilde{\lambda}^{T}$ has rank at most $r$ where $0\leq r\leq
k\leq n$. We wish to express this property in terms of the $k\times k$ minors
of the matrices $\lambda$ and $\widetilde{\lambda}$. This situation arises in
the study of scattering amplitudes in quantum field theory [3]. The special
case when $k=2$ and $r=0$ is known as spinor-helicity formalism; for textbook
basics see [3, Section 1.8] and [16, Section 2.2]. In physics, it is customary
to write $\langle ij\rangle$ for the $2\times 2$ minors of $\lambda$ and
$[ij]$ for the $2\times 2$ minors of $\widetilde{\lambda}$, where $1\leq
i<j\leq n$, and these minors satisfy the momentum conservation relations.

######  Example 1.1 ($k=2,n=5,r=0$).

We consider the two skew-symmetric $5\times 5$ matrices

$\\!\\!P\,=\,\small\begin{pmatrix}\,\,0&\langle 12\rangle&\langle
13\rangle&\langle 14\rangle&\langle 15\rangle\\\ \\!-\langle
12\rangle&0&\langle 23\rangle&\langle 24\rangle&\langle 25\rangle\\\
\\!-\langle 13\rangle&\\!\\!-\langle 23\rangle&0&\langle 34\rangle&\langle
35\rangle\\\ \\!-\langle 14\rangle&\\!\\!-\langle 24\rangle&\\!\\!-\langle
34\rangle&0&\langle 45\rangle\\\ \\!-\langle 15\rangle&\\!\\!-\langle
25\rangle&\\!\\!-\langle 35\rangle&\\!\\!-\langle 45\rangle&0\\\
\end{pmatrix}\quad{\rm and}\quad
Q\,=\,\small\begin{pmatrix}\,\,0&[12]&[13]&[14]&[15]\\\
\\!-[12]&0&[23]&[24]&[25]\\\ \\!-[13]&\\!\\!-[23]&0&[34]&[35]\\\
\\!-[14]&\\!\\!-[24]&\\!\\!-[34]&0&[45]\\\
\\!-[15]&\\!\\!-[25]&\\!\\!-[35]&\\!\\!-[45]&0\\\ \end{pmatrix}\\!.$

These matrices have rank two, meaning that the $4\times 4$ Pfaffians vanish
for both matrices:

$\\!\\!\\!\langle ij\rangle\langle kl\rangle-\langle ik\rangle\langle
jl\rangle+\langle il\rangle\langle
jk\rangle\,\,=\,\,[ij][kl]-[ik][jl]+[il][jk]\,=\,0\,\,\,\,\hbox{for}\,\,\,1\\!\leq\\!i\\!<\\!j\\!<\\!k\\!<\\!l\\!\leq\\!5.$
(1)

These quadratic Plücker relations are known as Schouten identities in physics
[3, eqn (1.116)]. Momentum conservation [3, eqn (1.117)] stipulates that the
product $P\cdot Q^{T}$ is the zero matrix:

$\langle i1\rangle[1j]+\langle i2\rangle[2j]+\langle i3\rangle[3j]+\langle
i4\rangle[4j]+\langle i5\rangle[5j]\,=\,0\quad\hbox{for}\,\,\,1\leq i,j\leq
5.$ (2)

In total, we have a system of $5+5+25=35$ quadratic equations in
$\binom{5}{2}+\binom{5}{2}=20$ unknowns. The equations (1) define a product of
two Grassmannians ${\rm Gr}(2,5)\times{\rm
Gr}(2,5)\subset\mathbb{P}^{9}\times\mathbb{P}^{9}$. Inside this product, the
bilinear equations (2) cut out a variety of dimension $8$. This is our spinor-
helicity variety, denoted ${\rm SH}(2,5,0)$. Its bidegree in
$\mathbb{P}^{9}\times\mathbb{P}^{9}$ is the cohomology class

$5s^{3}t^{7}+10s^{4}t^{6}+12s^{5}t^{5}+10s^{6}t^{4}+5s^{7}t^{3}\,\,\in\,\,H^{*}(\mathbb{P}^{9}\times\mathbb{P}^{9},\mathbb{Z}).$
(3)

The $35$ quadrics (1) and (2) generate the prime ideal of ${\rm SH}(2,5,0)$.
This ideal coincides with the ideal of the Grassmannian ${\rm Gr}(3,6)$, which
has codimension $10$ and degree $42$ in $\mathbb{P}^{19}$. This identification
was observed by Bossinger, Drummond and Glew [7], who used the term massless
scattering ideal and the notation $I_{\rm 5pt}$ for the ideal of ${\rm
SH}(2,5,0)$. In [7, Section 6.2] they derive the tropicalization of ${\rm
SH}(2,5,0)$ from that of ${\rm Gr}(3,6)$; see [22, Example 4.4.10].

The Grassmannian ${\rm Gr}(k,n)$ is the subvariety of
$\mathbb{P}^{\binom{n}{k}-1}$ defined by the Plücker equations. Points in
${\rm Gr}(k,n)$ represent $k$-dimensional linear subspaces in
$\mathbb{C}^{n}$. See e.g. [23, Chapter 5]. Every point in ${\rm Gr}(k,n)$ is
the row space of a matrix $\lambda\in\mathbb{C}^{k\times n}$ of rank $k$. We
consider the set

$\bigl{\\{}\,(\lambda,\widetilde{\lambda})\,\,:\,\,\lambda,\widetilde{\lambda}\in\mathbb{C}^{k\times
n}\,,\,\,{\rm rank}(\lambda)={\rm rank}(\widetilde{\lambda})=k,\,\,{\rm
and}\,\,\,{\rm rank}\bigl{(}\lambda\cdot\widetilde{\lambda}^{T}\bigr{)}\leq
r\,\bigr{\\}}.$ (4)

We define the spinor-helicity variety to be the image of (4) in the product of
Grassmannians:

${\rm SH}(k,n,r)\,\,\subset\,\,{\rm Gr}(k,n)\times{\rm
Gr}(k,n)\,\,\subset\,\,\mathbb{P}^{\binom{n}{k}-1}\times\mathbb{P}^{\binom{n}{k}-1}.$
(5)

In what follows, we study the algebra, combinatorics and geometry of the
inclusion (5). Our presentation aims to be accessible to a wide range of
readers, not just from mathematics, but also from physics. The prerequisites
are at the level of the algebra textbooks [23, 24, 27].

The motivation for this project arose from our desire to understand the
spinor-helicity formalism in physics. The variety ${\rm SH}(2,n,0)$ is widely
used for scattering amplitudes [3, 16]. Cachazo, Early, Guevara and Mizera
[11, Section 5.1] proposed the variety ${\rm SH}(k,n,k-2)$ as a model to
encode kinematic data for particle scattering. Scattering amplitudes in the
CEGM model are computed by integrating over the moduli space $X(k,n)$ of $n$
points in $\mathbb{P}^{k-1}$. The articles [1, 12, 28] studied the scattering
potential on $X(k,n)$. The nonlinear structure of the kinematic data was
highlighted in Lam’s lectures [21]. We here examine this in detail.

The kinematic data are summarized in the Mandelstam invariants (27). In the
$k=2$ case from Example 1.1, the moduli space is $X(2,n)=\mathcal{M}_{0,n}$,
and the Mandelstam invariants are

$s_{ij}\,\,=\,\,\langle ij\rangle[ij].$ (6)

These quantities play the role of the data in the log-likelihood
interpretation of [28]. Thus, from the algebraic statistics perspective, our
topic here is the geometry of data space. The fundamental object which
underlies this geometry is the spinor-helicity variety ${\rm SH}(k,n,r)$.

The article is organized as follows. In Section 2 we present quadratic
polynomials that form a Gröbner basis for the prime ideal of ${\rm
SH}(k,n,r)$. The underlying toric degeneration is represented by a poset
constructed from two copies of Young’s lattice for ${\rm Gr}(k,n)$. The
special case $r=0$ is understood by identifying ${\rm SH}(k,n,0)$ with the
two-step flag variety ${\rm Fl}(k,n-k;\mathbb{C}^{n})$. In Section 3 we
express the momentum conservation equations by a matrix product $PQ^{T}$ which
generalizes (2), and we show that these equations generate the prime ideal of
(5). Theorem 3.5 features a Khovanskii basis for the coordinate ring of ${\rm
SH}(k,n,r)$.

Section 4 investigates the polynomial relations among the the Mandelstam
invariants. These relations define the Mandelstam variety ${\rm M}(k,n,r)$ in
the kinematic subspace of $\mathbb{P}^{\binom{n}{k}-1}$. We study both
parametric and implicit representations of this variety. The generators of its
prime ideal for the case $k=2$ are presented in Theorem 4.5. We note in
Proposition 4.13 that ${\rm M}(k,n,k)$ is the Hadamard product (see [5]) of
the Grassmannian ${\rm Gr}(k,n)$ with itself.

In Section 5 we turn to positive geometry and tropical geometry. We introduce
the positive parts of ${\rm SH}(k,n,r)$ and ${\rm M}(k,n,r)$, we discuss their
boundaries, and we compute some associated tropical varieties. For $r=0$,
these structures arise from the flag variety.

In Section 6 we study the scattering correspondence. This is a variety in the
product space

${\rm M}(k,n,r)\,\times\,X(k,n),$

where $X(k,n)$ is the moduli space for $n$ points in $\mathbb{P}^{k-1}$. It
parametrizes pairs of Mandelstam invariants and solutions to their scattering
equations; see [21, eqn (0.2)]. This mirrors the likelihood correspondence in
statistics [19, Definition 1.5]. Building on [21, Section 4.3], we offer a
mathematical perspective on results from the physics literature, mostly for
$k=2$.

This article is accompanied by software and data. These materials are made
available in the MathRepo collection at MPI-MiS via
https://mathrepo.mis.mpg.de/SpinorHelicity.

##  2 Two Grassmannians and their Posets

We fix two copies of the Grassmannian ${\rm
Gr}(k,n)\subset\mathbb{P}^{\binom{n}{k}-1}$. The first Grassmannian has
Plücker coordinates $\langle i_{1}i_{2}\ldots i_{k}\rangle$, representing
maximal minors of $\lambda$. The second one has Plücker coordinates
$[i_{1}i_{2}\ldots i_{k}]$, representing maximal minors of
$\widetilde{\lambda}$. These expressions are antisymmetric, so we usually
assume $1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n$. For instance, for $k=3$,

$\begin{matrix}&\langle 123\rangle=-\langle 132\rangle=-\langle
213\rangle=\,\langle 231\rangle=\,\langle 312\rangle\,=-\langle 321\rangle\\\
{\rm
and}\quad&\,[123]\,=-[132]\,=\,-[213]\,\,=\,[231]\,=\,\,[312]\,=-[321].\end{matrix}$
(7)

Their relations are given by two copies of the Plücker ideal, denoted
$J_{k,n}$ and ${\widetilde{J}}_{k,n}$. To describe this ideal, we introduce
Young’s lattice $Y_{k,n}$. This is the partially ordered set (poset) whose
elements are the $\binom{n}{k}$ Plücker coordinates. The order relation in
$Y_{k,n}$ is defined by

$\langle i_{1}i_{2}\cdots i_{k}\rangle\,\leq\,\langle j_{1}j_{2}\cdots
j_{k}\rangle\quad:\Longleftrightarrow\quad i_{1}\leq j_{1}\,\,{\rm
and}\,\,i_{2}\leq j_{2}\,\,{\rm and}\,\,\cdots\,\,{\rm and}\,\,i_{k}\leq
j_{k}.$

Let $\widetilde{Y}_{k,n}$ be a second copy of Young’s poset, but now with the
order relation reversed:

$[i_{1}i_{2}\cdots i_{k}]\,\leq\,[j_{1}j_{2}\cdots
j_{k}]\quad:\Longleftrightarrow\quad i_{1}\geq j_{1}\,\,{\rm and}\,\,i_{2}\geq
j_{2}\,\,{\rm and}\,\,\cdots\,\,{\rm and}\,\,i_{k}\geq j_{k}.$

The following result on the ideal $J_{k,n}$ of the Grassmannian is well-known
(see [27, §3.1]).

######  Proposition 2.1.

The prime ideal $J_{k,n}$ is generated by the quadratic Plücker relations

$\sum_{s=0}^{k}\,(-1)^{s}\cdot\langle\,i_{1}\,i_{2}\,\cdots\,i_{k-1}\,j_{s}\,\rangle\cdot\langle
j_{0}j_{1}\cdots j_{s-1}j_{s+1}\cdots j_{k}\rangle.$ (8)

These quadrics are a Gröbner basis for the reverse lexicographic term order
given by any linear extension of $Y_{k,n}$. The initial ideal of $J_{k,n}$ is
generated by the incomparable pairs in $Y_{k,n}$.

The key point of this result is that every incomparable pair lifts to a
quadric in $J_{k,n}$.

######  Example 2.2 ($k=4,n=8$).

The elements $\langle 1278\rangle$ and $\langle 3456\rangle$ are incomparable
in $Y_{4,8}$. The Plücker relation (8) for the triple $i_{1}i_{2}i_{3}=127$
and the quintuple $j_{0}j_{1}j_{2}j_{3}j_{4}=34568$ is

${\bf\langle 1278\rangle\langle 3456\rangle}\,+\,\langle 1267\rangle\langle
3458\rangle-\langle 1257\rangle\langle 3468\rangle+\langle 1247\rangle\langle
3568\rangle-\langle 1237\rangle\langle 4568\rangle.$

The monomials are listed in the term order we have chosen. The initial
monomial is the prescribed incomparable pair. Our quadric is not in the
reduced Gröbner basis since it has incomparable trailing terms. The
corresponding element in the reduced Gröbner basis equals

$\begin{matrix}{\bf\langle 1278\rangle\langle 3456\rangle-\langle
1256\rangle\langle 3478\rangle}+\langle 1246\rangle\langle 3578\rangle-\langle
1245\rangle\langle 3678\rangle\quad\qquad\\\ \qquad\qquad\qquad-\,\langle
1236\rangle\langle 4578\rangle\,+\,\langle 1235\rangle\langle
4678\rangle\,-\,\langle 1234\rangle\langle 5678\rangle.\end{matrix}$

That quadric has the virtue that its initial binomial $\langle
1278\rangle\langle 3456\rangle-\langle 1256\rangle\langle 3478\rangle$ is
consistent with the toric degeneration of the Grassmannian ${\rm Gr}(4,8)$
given by Young’s lattice $Y_{4,8}$. Indeed, for the incomparable pair $\langle
1278\rangle\langle 3456\rangle$, the meet is $\langle 3478\rangle$ and the
join is $\langle 1256\rangle$. Algebraically, this is the Khovanskii basis (or
SAGBI basis) structure in [27, Theorem 3.2.9].

######  Corollary 2.3.

The number of generators for $J_{k,n}$, or of incomparable pairs in $Y_{k,n}$,
equals

$\frac{1}{2}\left[\binom{n}{k}+1\right]\binom{n}{k}\,\,-\,\,\frac{(n+1)(n-k+1)}{k+1}\prod_{i=0}^{k-2}\frac{(n-i)^{2}}{(k-i)^{2}}.$
(9)

###### Proof.

The first term is the number of all quadratic monomials in the $\binom{n}{k}$
variables $\langle i_{1}i_{2}\cdots i_{k}\rangle$. From this we subtract the
number of standard monomials, which is the number of semi-standard Young
tableaux of shape $k\times 2$ with fillings in $\\{1,2,\ldots,n\\}$. That
number is given by the hook-content formula from combinatorics, which we made
explicit in (9). ∎

We now turn to the spinor-helicity variety. Rephrasing the definition in
(4)-(5), this is

$\mathrm{SH}(k,n,r)\,\,=\,\,\bigl{\\{}\,(V,W)\in\mathrm{Gr}(k,n)\times\mathrm{Gr}(k,n)\,\,\colon\dim(V\cap
W^{\perp})\geq k-r\,\bigr{\\}},$ (10)

where $W^{\perp}$ is the space orthogonal to $W$ with respect to the standard
inner product on $\mathbb{C}^{n}$. In the setting of the Introduction, $V$ and
$W$ are the row spaces of $\lambda$ and $\widetilde{\lambda}$ respectively.

######  Remark 2.4 (Involution).

There is a canonical involution on $\mathrm{SH}(k,n,r)$, defined by swapping
the subspaces $V$ and $W$. This interchanges the coordinates $\langle
i_{1}i_{2}\cdots i_{k}\rangle$ and $[i_{1}i_{2}\cdots i_{k}]$.

######  Proposition 2.5.

Fix integers $k,n,r$ such that $0\leq r\leq k$ and $2k\leq r+n$. The spinor-
helicity variety $\,\mathrm{SH}(k,n,r)\,$ is non-empty and irreducible in
$\mathbb{P}^{\binom{n}{k}-1}\times\mathbb{P}^{\binom{n}{k}-1}$. Its dimension
equals

$\dim(\mathrm{SH}(k,n,r))\,\,=\,\,2k(n-k)-(k-r)^{2}.$ (11)

If $r=0$ then it is linearly isomorphic to the two-step flag variety $\,{\rm
Fl}(k,n-k;\mathbb{C}^{n})$.

###### Proof.

Our hypothesis that $k={\rm dim}(V)$ and $n-k={\rm dim}(W^{\perp})$ are at
least $k-r$ is necessary for $\mathrm{SH}(k,n,r)$ to be non-empty. We assume
$k-r\geq 0$ to rule out trivial cases. The projection of $\mathrm{SH}(k,n,r)$
onto the first factor equals ${\rm Gr}(k,n)$, which is irreducible of
dimension $k(n-k)$. The fibers are subvarieties in the second factor ${\rm
Gr}(k,n)$. In local affine coordinates, the fiber over $V$ consists of all
$k\times(n-k)$ matrices of rank at most $r$. This is an irreducible variety of
codimension $(k-r)^{2}$. Hence, each fiber is irreducible of the same
dimension $k(n-k)-(k-r)^{2}$. From this we conclude that $\mathrm{SH}(k,n,r)$
is irreducible of dimension $2k(n-k)-(k-r)^{2}$.

Fix $r=0$ and recall that $k\leq n-k$. By passing from $W$ to $W^{\perp}$, we
can view ${\rm SH}(k,n,0)$ as a subvariety in ${\rm Gr}(k,n)\times{\rm
Gr}(n-k,n)$. Its points are pairs $(V,W^{\perp})$ of linear subspaces in
$\mathbb{C}^{n}$ such that $V\subseteq W^{\perp}$. In other words, its points
are two-step flags in $\mathbb{C}^{n}$. Hence ${\rm SH}(k,n,0)$ coincides with
the flag variety ${\rm Fl}(k,n-k;\mathbb{C}^{n})$ in its Plücker embedding in
$\,\mathbb{P}^{\binom{n}{k}-1}\times\,\mathbb{P}^{\binom{n}{k}-1}$. ∎

Let $R=\mathbb{C}\bigl{[}\langle i_{1}\dots i_{k}\rangle,[j_{1}\dots
j_{k}]\bigr{]}$ be the polynomial ring in the $2\binom{n}{k}$ bracket
variables. Let $S=\mathbb{C}[\bf{x}]$ be the polynomial ring in the entries of
an $(n-k+r)\times n$ matrix ${\bf x}=(x_{ij})$. We write
$\phi_{k,n,r}:R\rightarrow S$ for the homomorphism which maps $\langle
I\rangle=\langle i_{1}i_{2}\cdots i_{k}\rangle$ to the $k\times k$ minor of
${\bf x}$ in the rows $1,2,\ldots,k$ and columns $I$, and which maps
$[J]=[j_{1}j_{2}\cdots j_{k}]$ to $(-1)^{j_{1}+j_{2}+\cdots+j_{k}}$ times the
$(n\\!-\\!k)\times(n\\!-\\!k)$ minor of ${\bf x}$ in the rows
$r+1,\ldots,n-k+r$ and columns $[n]\backslash J$. Note that all such minors
involve the $k-r$ middle rows, which are indexed by $r+1,\ldots,k$.

######  Remark 2.6 (Parametrization).

Let $I_{k,n,r}$ denote the kernel of the ring map $\phi_{k,n,r}$. This kernel
is a homogeneous prime ideal in $R$, and its zero set is the spinor-helicity
variety ${\rm SH}(k,n,r)$. Hence, the subalgebra $\phi_{k,n,r}(R)$ of
$S=\mathbb{C}[{\bf x}]$ is isomorphic to the coordinate ring of ${\rm
SH}(k,n,r)$. Indeed, following (10), the minors $\phi_{k,n,r}(\langle
I\rangle)$ are the Plücker coordinates for the space $V$, while the signed
minors $\phi_{k,n,r}([J])$ are the Plücker coordinates for $W^{\perp}$. The
condition ${\rm dim}(V\cap W^{\perp})\geq k-r$ is encoded by the overlap in
the $k-r$ middle rows of ${\bf x}$.

We will describe a Gröbner basis of quadrics for $I_{k,n,r}$. The initial
monomials admit a combinatorial description which extends that for
Grassmannians given in Proposition 2.1. We start out with our two copies of
Young’s lattice, $Y_{k,n}$ and $\widetilde{Y}_{k,n}$. We define a new poset
$\mathcal{P}_{k,n,r}$ as follows. As a set, $\mathcal{P}_{k,n,r}$ is the
disjoint union of $Y_{k,n}$ and $\widetilde{Y}_{k,n}$. All order relations in
$Y_{k,n}$ and $\widetilde{Y}_{k,n}$ remain order relations in
$\mathcal{P}_{k,n,r}$. In addition, there are $\binom{2k-2r}{k-r}$ covering
relations

$[12\cdots r\,i_{r+1}\cdots i_{k}]\,\leq\,\langle 12\cdots r\,j_{r+1}\cdots
j_{k}\rangle,$ (12)

one for each ordered set partition
$\,\\{r+1,r+2,\ldots,2k-r\\}=\\{i_{r+1},\ldots,i_{k}\\}\sqcup\\{j_{r+1},\ldots,j_{k}\\}$.
The poset $\mathcal{P}_{k,n,r}$ is the transitive closure of these relations.
Note that $\mathcal{P}_{k,n,r}$ is a graded poset, with unique minimal element
$[n{-}k{+}1\,\cdots\,n{-}1\,n]$ and unique maximal element $\langle
n{-}k{+}1\,\cdots\,n{-}1\,n\rangle$. The Hasse diagram of
$\mathcal{P}_{k,n,r}$ is shown in Figure 1 for $k=2,n=6,r=0$.

$\langle 56\rangle$$\langle 46\rangle$$\langle 36\rangle$$\langle
26\rangle$$\langle 16\rangle$$\langle 45\rangle$$\langle 35\rangle$$\langle
25\rangle$$\langle 15\rangle$$\langle 34\rangle$$\langle 24\rangle$$\langle
14\rangle$$\langle 23\rangle$$\langle 13\rangle$$\langle
12\rangle$[56][46][36][26][16][45][35][25][15][34][24][14][23][13][12]

Figure 1: The poset $\mathcal{P}_{2,6,0}$ is created from $Y_{2,6}$ and
$\widetilde{Y}_{2,6}$ by adding six covering relations.

We are now prepared to state our first theorem on the spinor-helicity variety
${\rm SH}(k,n,r)$.

######  Theorem 2.7.

The prime ideal $I_{k,n,r}$ is minimally generated by quadratic forms. These
quadrics are a Gröbner basis for the reverse lexicographic term order given by
any linear extension of $\mathcal{P}_{k,n,r}$. The initial ideal of
$I_{k,n,r}$ is generated by the incomparable pairs in $\mathcal{P}_{k,n,r}$.

###### Proof.

We first assume $r=0$. By Proposition 2.5, our ideal $I_{k,n,0}$ is the ideal
of the two-step flag variety ${\rm Fl}(k,n-k;\mathbb{C}^{n})$. The quadratic
Gröbner basis for that ideal is derived from the well-known straightening law
for flag varieties. We refer to [24, Chapter 14] for a textbook exposition.
That exposition emphasizes the case of the complete flag variety. This case
applies to our situation as follows. Let $\mathcal{P}$ be the poset on all
$2^{n}$ subsets of $\\{1,2,\ldots,n\\}$ that was introduced in [24, Section
14.2]. The restriction of that poset to subsets that have size $k$ or $n-k$ is
isomorphic to our poset $\mathcal{P}_{k,n,0}$. The poset isomorphism maps
$(n-k)$-sets to their complements. With this, our assertion for $r=0$ follows
from [24, Theorem 14.6].

We next present the proof for $r\geq 1$. This will generalize the known
construction we used for $r=0$. We consider the skew Young diagram
$\lambda/\mu$ where $\lambda=(n-k+r,r)$ and $\mu=(r)$. A filling of
$\lambda/\mu$ with entries in $[n]=\\{1,2,\ldots,n\\}$ is assumed to have its
rows strictly increasing. Hence there are $\binom{n}{k}^{2}$ such fillings. A
filling is semi-standard if the $k-r$ non-trivial columns are weakly
increasing. If this is not the case then the filling of $\lambda/\mu$ is
called non-standard.

With these definitions in place, our poset admits the following alternative
description:

$\begin{matrix}&\quad\langle i_{1}i_{2}\,\cdots
i_{k}\,\rangle\,\geq\,[j_{1}j_{2}\,\cdots\,j_{k}]\qquad\qquad\hbox{holds in
$\mathcal{P}_{k,n,r}$,}\\\ \iff&\langle
i_{r+1}\\!-\\!r\,\cdots\,i_{k}\\!-\\!r\rangle\,\geq\,[j_{r+1}\\!-\\!r\,\cdots\,j_{k}\\!-\\!r]\quad\hbox{holds
in $\mathcal{P}_{k-r,n-r,0}$,}\\\ \iff&\hbox{the filling of $\lambda/\mu$ with
$[n]\backslash\\{j_{1},\ldots,j_{k}\\}$ and $\\{i_{1},\ldots,i_{k}\\}$ is
semi-standard.}\end{matrix}$ (13)

The proof of our theorem is now analogous to that of [24, Theorem 14.6]. Fix
an incomparable pair $\langle I\rangle[J]$ in the poset $\mathcal{P}_{k,n,r}$,
and consider the corresponding non-standard skew tableaux

$\lambda/\mu\quad=\qquad\setcounter{MaxMatrixCols}{11}\begin{bmatrix}&&&j_{1}^{\prime}&\cdots&j^{\prime}_{l}&\cdots&j^{\prime}_{k-r}&j^{\prime}_{k-r+1}&\cdots&j^{\prime}_{n-k}\\\
i_{1}&\cdots&i_{r}&i_{r+1}&\cdots&i_{r+l}&\cdots&i_{k}&&&\\\ \end{bmatrix}.$
(14)

The rows are increasing, and $i_{r+l}<j^{\prime}_{l}$ is the leftmost
violation, and $\\{j^{\prime}_{1},\ldots,j^{\prime}_{n-k}\\}=[n]\backslash J$.

By summing over all permutations $\pi$ of
$i_{1}<\cdots<i_{r+l}<j^{\prime}_{l}<\cdots<j^{\prime}_{n-k}$, we obtain

$\sum_{\pi}{\rm sign}(\pi)\cdot\langle\pi(I)\rangle\cdot[\pi([n]\backslash
J)]\quad\in\,\,R.$ (15)

This is the analogue to [24, eqn (14.2)]. The image of (15) under
$\phi_{k,n,r}$ is an alternating multilinear form in $n-k+r+1$ column vectors
of the matrix ${\bf x}$. But, the matrix has only $n-k+r$ rows, so this is the
zero polynomial. Therefore (15) lies in $I_{k,n,r}={\rm
kernel}(\phi_{k,n,r})$. Finally, we note that the initial monomial of (15) is
the monomial $\langle I\rangle[J]$ we started out with.

To complete the proof, we need to show that the semi-standard monomials in $R$
are linearly independent modulo $I_{k,n,r}$. The argument for this follows
that in the proof of [24, Theorem 14.6]. Namely, we consider any monomial in
$S$ and we write it as in [24, eqn (14.4)]. There exists a unique semi-
standard skew tableau whose image under $\phi_{k,n,r}$ has that initial
monomial. Thus, no cancellation is possible, and this finishes the proof of
Theorem 2.7. ∎

######  Example 2.8 ($k=2,n=6,r=0$).

The ideal $I_{2,6,0}$ is generated by $15+15+36=66$ quadrics which form a
Gröbner basis. Their initial monomials are the incomparable pairs in the poset
$\mathcal{P}_{2,6,0}$ which is shown in Figure 1. The $15$ initial monomials
from $J_{2,6}$ are the pairs $\langle ij\rangle\langle kl\rangle$ in
$Y_{2,6}$. The $15$ initial monomials from $\widetilde{J}_{2,6}$ are the pairs
$[ij][kl]$ in $\widetilde{Y}_{2,6}$. Finally, there are $36$ mixed initial
monomials $\langle ij\rangle[kl]$, corresponding to bilinear generators of
$I_{2,6,0}$.

The poset $\mathcal{P}_{2,5,0}$ arises in Figure 1 from deleting the upper rim
$\langle 16\rangle,\langle 26\rangle,\langle 36\rangle,\langle
46\rangle,\langle 56\rangle$ and the lower rim $[16],[26],[36],[46],[56]$.
Thus $\mathcal{P}_{2,5,0}$ has $\binom{5}{2}+\binom{5}{2}=20$ elements. It has
five incomparable pairs $\langle ij\rangle\langle kl\rangle$, five
incomparable pairs $[ij][kl]$, and $25$ mixed incomparable pairs $\langle
ij\rangle[kl]$. These are the initial monomials of the $35$ ideal generators
in Example 1.1.

The following result concerns the number of incomparable pairs in the poset
$\mathcal{P}_{k,n,r}$.

######  Lemma 2.9.

The number of mixed incomparable pairs $\langle
i_{1},\dots,i_{k}\rangle[j_{1},\dots,j_{k}]$ equals $\binom{n}{k-r-1}^{2}$.

###### Proof.

Let $\lambda=(n-k+r,\ k)$ and $\mu=(r)$. The generating function for semi-
standard skew tableaux of shape $\lambda/\mu$ is the skew Schur polynomial
$s_{\lambda/\mu}$, which can be written as

$s_{\lambda/\mu}\,\,=\,\,\sum_{\nu}c_{\mu,\nu}^{\lambda}s_{\nu}.$

Here $c^{\lambda}_{\mu,\nu}$ are the Littlewood-Richardson coefficients. In
our special case $\mu=(r)$, Pieri’s rule tells us that
$c^{\lambda}_{\mu,\nu}=1$ if $\nu$ is obtained from $\lambda$ by removing $r$
boxes and $0$ otherwise. Using the hook-content formula to evaluate each
individual $s_{\nu}(1,\dots,1)$, we can now compute the number of semi-
standard skew tableaux of shape $\lambda/\mu$ with fillings in $[n]$. That
number equals

$\begin{matrix}s_{\lambda/\mu}(1,1,\ldots,1)\quad\,&=&\,\,\sum_{\begin{subarray}{c}\nu\text{
obtained from $\lambda$ }\\\ \text{by removing $r$
boxes}\end{subarray}}s_{\nu}(1,1,\dots,1)\vskip 6.0pt plus 2.0pt minus
2.0pt\\\ &=&\quad\sum_{\ell=0}^{r}s_{(n-k+\ell,k-\ell)}(1,1,\dots,1)\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&=&\quad\sum_{\ell=0}^{r}\left[\binom{n}{k-\ell}^{2}-\binom{n}{k-\ell-1}^{2}\right]\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
&=&\quad\binom{n}{k}^{\\!2}-\,\binom{n}{k-r-1}^{\\!2}.\end{matrix}$ (16)

Since $\binom{n}{k}^{2}$ is the number of all tableaux, we see that the number
of non-standard skew tableau of shape $\lambda/\mu$ equals
$\binom{n}{k-r-1}^{2}$. The equivalence in (13) now completes the proof. ∎

Theorem 2.7 and Lemma 2.9 imply the following result about the spinor-helicity
variety.

######  Corollary 2.10.

The number of minimal generators of $I_{k,n,r}$ equals twice (9) plus
$\binom{n}{k-r-1}^{2}$. These generators are quadratic and we can arrange them
to form a reduced Gröbner basis.

After its dimension, the second-most important invariant of a variety in a
projective space is its degree. For a variety in a product of two projective
spaces, one considers the bidegree, which is a homogeneous polynomial in two
variables $s$ and $t$. The degree of that polynomial is the codimension of the
variety. We saw an example in (3). By definition, the bidegree of ${\rm
SH}(k,n,r)$ is its class in the cohomology ring of ambient product of
projective spaces:

$H^{*}\left(\mathbb{P}^{\binom{n}{k}-1}\times\mathbb{P}^{\binom{n}{k}-1},\mathbb{Z}\right)\,\,=\,\,\mathbb{Z}[s,t]\,/\bigl{\langle}\,s^{\binom{n}{k}},\,t^{\binom{n}{k}}\,\bigr{\rangle}.$

We now present a general formula for the cohomology class of the spinor-
helicity variety.

######  Corollary 2.11.

The bidegree of $\,{\rm SH}(k,n,r)$ is equal to

$(st)^{\binom{n}{k}-k(n-k)-1}\cdot\sum c(i_{1}i_{2}\cdots i_{k})\cdot
c(j_{1}j_{2}\cdots j_{k})\cdot
s^{i_{1}+i_{2}+\cdots+i_{k}-\binom{k+1}{2}}\cdot
t^{j_{1}+j_{2}+\cdots+j_{k}-\binom{k+1}{2}},$ (17)

where we sum over all covering relations in (12), and $c(i_{1}i_{2}\cdots
i_{k})$ denotes the number of maximal chains from $\langle i_{1}i_{2}\cdots
i_{k}\rangle$ to the top element $\langle n{-}k{+}1\,\cdots\,n{-}1\,n\rangle$
in Young’s lattice $Y_{k,n}$.

###### Proof.

The bidegree of $I_{k,n,r}$ equals the bidegree of the initial monomial ideal
${\rm in}(I_{k,n,r})$. The latter is generated by the incomparable pairs in
$\mathcal{P}_{k,n,r}$. The bidegree is the multidegree of [24, §8.5] for the
$\mathbb{Z}^{2}$-grading at hand. It is additive over top-dimensional primary
components, and we can use the formula in [24, Theorem 8.44] for its
evaluation. The associated primes of ${\rm in}(I_{k,n,r})$ correspond to the
maximal chains of $\mathcal{P}_{k,n,r}$. Each maximal chain starts out at the
bottom of $\widetilde{Y}_{k,n}$, it uses precisely one of the covering
relations in (12) to transition from $\widetilde{Y}_{k,n}$ to $Y_{k,n}$, and
it then proceeds to the top of $Y_{k,n}$. Thus there are precisely
$c(i_{1}i_{2}\cdots i_{k})\cdot c(j_{1}j_{2}\cdots j_{k})$ maximal chains
which use the specific covering relation in (12). The associated monomial in
$s$ and $t$ records the height at which the transition from
$\widetilde{Y}_{k,n}$ to $Y_{k,n}$ in $\mathcal{P}_{k,n,r}$ takes place. ∎

######  Example 2.12 ($k=2,n=6,r=0$).

The bidegree of $I_{2,6,0}$ is a sum of monomials $s^{i}t^{j}$ over the
maximal chains of the poset $\mathcal{P}_{2,6,0}$ in Figure 1. The degree
$i+j=16$ of each monomial is the codimension of ${\rm SH}(2,6,0)$ in
$\mathbb{P}^{14}\times\mathbb{P}^{14}$. Counting paths in Young’s lattice
$Y_{2,6}$, we see

$c(12)=14,\,c(13)=14,\,c(14)=9,\,c(23)=5,\,c(24)=5,\,c(34)=2.$

The bidegree of $I_{2,6,0}$ is the polynomial
$\,28s^{6}t^{10}+70s^{7}t^{9}+90s^{8}s^{8}+70s^{9}t^{7}+28s^{10}t^{6}$

$=\,c(12)c(34)s^{6}t^{10}+c(13)c(24)s^{7}t^{9}+2c(14)c(23)s^{8}s^{8}+c(24)c(13)s^{9}t^{7}+c(34)c(12)s^{10}t^{6}.$

Hence total number of maximal chains in $\mathcal{P}_{2,6,0}$ equals
$28+70+90+70+28=286$.

######  Example 2.13 ($k=3,n=7,r=1$).

The poset $\mathcal{P}_{3,7,1}$ has $70$ elements and $312816$ maximal chains.
It arises from $Y_{3,7}$ and $\widetilde{Y}_{3,7}$ by adding six covering
relations, as shown in Figure 2. The ideal $I_{3,7,1}$ has $140+140+49=329$
minimal generators, one for each incomparable pair; by Corollary 2.10. The
blue numbers $c(ijk)$ count maximal chains from $\langle ijk\rangle$ to
$\langle 567\rangle$ in $Y_{3,7}$ or maximal chains from $[567]$ to $[ijk]$ in
$\widetilde{Y}_{3,7}$. By Corollary 2.11, the bidegree of $I_{3,7,1}$ equals
$(st)^{22}\bigl{(}462\cdot 56\,s^{4}\,+\,462\cdot 168\,s^{3}t\,+\,(252\cdot
210+210\cdot 252)\,s^{2}t^{2}\,+\,168\cdot 462\,st^{3}\,+\,56\cdot
462\,t^{4}\bigr{)}$.

$\langle 167\rangle$1$\langle 157\rangle$4$\langle 147\rangle$9$\langle
137\rangle$14$\langle 127\rangle$14$\langle 156\rangle$10$\langle
146\rangle$35$\langle 136\rangle$70$\langle 126\rangle$84$\langle
145\rangle$56$\langle 135\rangle$168$\langle 125\rangle$252$\langle
134\rangle$210$\langle 124\rangle$462$\langle 123\rangle$462$\langle
267\rangle$1$\langle 257\rangle$3$\langle 247\rangle$5$\langle
237\rangle$5$\langle 256\rangle$6$\langle 246\rangle$16$\langle
236\rangle$21$\langle 245\rangle$21$\langle 235\rangle$42$\langle
234\rangle$42$\langle 367\rangle$1$\langle 357\rangle$2$\langle
347\rangle$2$\langle 356\rangle$3$\langle 346\rangle$5$\langle
345\rangle$5$\langle 467\rangle$1$\langle 457\rangle$1$\langle
456\rangle$1$\langle
567\rangle$1$[167]$1$[157]$4$[147]$9$[137]$14$[127]$14$[156]$10$[146]$35$[136]$70$[126]$84$[145]$56$[135]$168$[125]$252$[134]$210$[124]$462$[123]$462$[267]$1$[257]$3$[247]$5$[237]$5$[256]$6$[246]$16$[236]$21$[245]$21$[235]$42$[234]$42$[367]$1$[357]$2$[347]$2$[356]$3$[346]$5$[345]$5$[467]$1$[457]$1$[456]$1$[567]$1

Figure 2:  The poset $\mathcal{P}_{3,7,1}$ governs the combinatorics of the
variety ${\rm SH}(3,7,1)\subset\mathbb{P}^{34}\times\mathbb{P}^{34}$.

##  3 Bilinear Relations and Khovanskii Basis

In Section 2 we took a route into the combinatorial commutative algebra of the
spinor-helicity variety. This journey continues in this section. We begin by
taking a closer look at the bilinear equations in $I_{k,n,r}$. Thereafter, we
introduce a toric degeneration of ${\rm SH}(k,n,r)$, based on the poset
$\mathcal{P}_{k,n,r}$. The resulting Khovanskii basis matches our earlier
Gröbner basis.

We now introduce two matrices $P$ and $Q$ whose rows are indexed by
$\binom{[n]}{k-r-1}$ and whose columns are indexed by $\binom{[n]}{r+1}$. Here
$[n]=\\{1,2,\ldots,n\\}$ and $\binom{[n]}{s}$ is the set of subsets of size
$s$ in $[n]$. The entries of our matrices are given by concatenating row
labels and column labels

$P_{I,J}\,=\,\langle\,I\,J\,\rangle\quad\text{and }\quad
Q_{I,J}\,=\,[\,I\,J\,]\qquad\hbox{for $\,I\in\binom{[n]}{k-r-1}\,$ and
$\,J\in\binom{[n]}{r+1}$.}$

Here $I$ and $J$ are increasing sequences which we concatenate. We pass to the
sorted Plücker coordinates $\,\langle I\,\cup\,J\rangle$ and $\,[I\,\cup\,J]$
from Section 2 by multiplying with $-1,+1$ or $0$, as in (7). In particular,
this means that $\,\langle I\,J\rangle\,=\,0\,$ and $\,[I\,J]\,=\,0\,$
whenever $I\cap J\not=\emptyset$.

######  Example 3.1 ($k=3,n=5,r=1$).

The matrix $P$ is square of format $10\times 10$. The rows and columns of $P$
are labeled by $12,13,14,15,23,24,25,34,35,45$, in this order. We find

$P\,=\,\small\begin{pmatrix}0&0&0&0&0&0&0&\langle 1234\rangle&\langle
1235\rangle&\langle 1245\rangle\\\ 0&0&0&0&0&\langle 1324\rangle&\langle
1345\rangle&0&0&\langle 1345\rangle\\\ 0&0&0&0&\langle 1423\rangle&0&\langle
1425\rangle&0&\langle 1435\rangle&0\\\ 0&0&0&0&\langle 1523\rangle&\langle
1524\rangle&0&\langle 1534\rangle&0&0\\\ 0&0&\langle 2314\rangle&\langle
2315\rangle&0&0&0&0&0&\langle 2345\rangle\\\ 0&\langle 2413\rangle&0&\langle
2415\rangle&0&0&0&0&\langle 2435\rangle&0\\\ 0&\langle 2513\rangle&\langle
2514\rangle&0&0&0&0&\langle 2534\rangle&0&0\\\ \langle 3412\rangle&0&0&\langle
3415\rangle&0&0&\langle 3425\rangle&0&0&0\\\ \langle 3512\rangle&0&\langle
3514\rangle&0&0&\langle 3524\rangle&0&0&0&0\\\ \langle 4512\rangle&\langle
4513\rangle&0&0&\langle 4523\rangle&0&0&0&0&0\\\ \end{pmatrix}\\!.$

Each entry is now replaced by Plücker coordinates with increasing indices. For
instance, we replace $\langle 1324\rangle$ by $-\langle 1234\rangle$. The
matrix $Q$ is identical to $P$ but with square brackets $[ijkl]$.

######  Example 3.2 ($k=2,r=0$).

This is the case of most interest in physics. Here $P$ and $Q$ are the skew-
symmetric $n\times n$ matrices of Plücker coordinates, shown for $n=5$ in
Example 1.1.

######  Example 3.3 ($r=k-1$).

The variety ${\rm SH}(k,n,k-1)$ is a hypersurface in ${\rm Gr}(k,n)\times{\rm
Gr}(k,n)$. Here $P$ and $Q$ are the row vectors of length $\binom{n}{k}$ whose
entries are the Plücker coordinates. The defining equation of this
hypersurface is the inner product of the two Plücker vectors:

$PQ^{T}\,=\,\sum\langle i_{1}i_{2}\cdots i_{k}\rangle[i_{1}i_{2}\cdots
i_{k}]\,\,=\,\,0.$

These examples guide us towards our next theorem, which is the main result in
Section 3.

######  Theorem 3.4.

The entries of the matrix $\,PQ^{T}$ generate the prime ideal of the spinor-
helicity variety ${\rm SH}(k,n,r)$ in the coordinate ring of $\,{\rm
Gr}(k,n)\times{\rm Gr}(k,n)$. In symbols, we have

$I_{k,n,r}\,\,=\,\,J_{n,k}+\widetilde{J}_{k,n}+\langle\text{entries of
}PQ^{T}\rangle.$ (18)

###### Proof.

We first show that the entries of $PQ^{T}$ vanish on the spinor-helicity
variety. Fix any point in ${\rm SH}(k,n,r)$, represented by a pair of $k\times
n$ matrices $\lambda$ and $\widetilde{\lambda}$ such that
$\lambda\cdot\widetilde{\lambda}^{T}$ has rank $\leq r$. By passing to the
$(r+1)$-st exterior power, we find that
$\,\wedge_{r+1}\lambda\cdot(\wedge_{r+1}\widetilde{\lambda})^{T}\,$ is the
zero matrix of format $\binom{k}{r+1}\times\binom{k}{r+1}$. In other words,
the row spaces of the $\binom{k}{r+1}\times\binom{n}{r+1}$ matrices
$\wedge_{r+1}\lambda$ and $\wedge_{r+1}\widetilde{\lambda}$ are orthogonal to
each other. The row vectors of the matrices $P$ and $Q$ are elements in these
row spaces. Therefore $P\cdot Q^{T}=0$ holds for our point
$(\lambda,\widetilde{\lambda})$.

The previous paragraph shows that the right hand side of (18) is contained in
the left hand side. Both ideals are generated by quadrics, and they contain
the Plücker ideals $J_{k,n}$ and $\widetilde{J}_{k,n}$. It therefore suffices
to show that the entries of $PQ^{T}$ span the space of all bilinear quadrics
in the ideal $I_{k,n,r}$. We know from Lemma 2.9 and Corollary 2.10 that this
space has dimension $\binom{n}{k-r-1}^{2}$. This number coincides with the
number of entries in the square matrix $PQ^{T}$. It therefore suffices to show
that the entries of $PQ^{T}$ are linearly independent over $\mathbb{Q}$.

We shall prove this by contradiction. The entry of $PQ^{T}$ in row $I$ and
column $J$ equals

$f_{IJ}\,\,\,=\sum_{L\in\binom{[n]}{r+1}}\epsilon_{I,L}\,\epsilon_{J,L}\
\langle IL\rangle[JL],$

where $\epsilon_{I,L}=\pm 1$ is the sign of the permutation that sorts the
string $IL$. Suppose that

$\sum_{I,J\in\binom{[n]}{k-r-1}}\\!\\!\\!\\!\alpha_{I,J}\cdot
f_{IJ}\,\,=\,\,0\qquad\hbox{for some scalars $\alpha_{IJ}\in\mathbb{Q}$.}$

We must show that each $\alpha_{IJ}$ is zero. The previous equation can be
rewritten as follows:

$\sum_{\begin{subarray}{c}I^{\prime},J^{\prime}\in\binom{[n]}{k}\\\
|I^{\prime}\cap J^{\prime}|\geq
r+1\end{subarray}}\biggl{[}\,\,\sum_{\begin{subarray}{c}L\subset
I^{\prime}\cap J^{\prime}\\\
|L|=r+1\end{subarray}}\\!\\!\epsilon_{I^{\prime}\setminus
L,L}\,\epsilon_{J^{\prime}\setminus L,L}\ \alpha_{I^{\prime}\setminus
L,J^{\prime}\setminus
L}\,\biggr{]}\,\langle\,I^{\prime}\,\rangle\,[\,J^{\prime}\,]\,\,\,=\,\,\,0.$

From this we conclude that, for any two $k$-subsets $I^{\prime},J^{\prime}$
with $|I^{\prime}\cap J^{\prime}|\geq r+1$, we have

$\begin{matrix}\qquad\sum_{\begin{subarray}{c}L\in\binom{I^{\prime}\cap
J^{\prime}}{r+1}\end{subarray}}\epsilon_{I^{\prime}\setminus
L,L}\,\epsilon_{J^{\prime}\setminus L,L}\ \alpha_{I^{\prime}\setminus
L,J^{\prime}\setminus L}\,\,=\,\,0\qquad\hbox{for
all}\,\,\,I^{\prime},J^{\prime}\in\binom{[n]}{k}.\end{matrix}$ (19)

Our goal is to show that all $\alpha$’s are zero. First consider the case
$I^{\prime}=J^{\prime}$. Here (19) reads

$\sum_{L\subset I^{\prime},|L|=r+1}\\!\\!\\!\\!\alpha_{I^{\prime}\setminus
L,I^{\prime}\setminus L}\,\,=\,\,0.$

We write these equations in the form $Ba=0$ where $a_{I}=\alpha_{I,I}$ and $B$
is an $\binom{n}{k}\times\binom{n}{k-r-1}$ matrix with entries in $\\{0,1\\}$.
The row indices are subsets $J\in\binom{[n]}{k}$ and the column indices are
subsets $I\subset\binom{[n]}{k-r-1}$. The matrix entry $B_{J,I}$ equals $1$
when $I\subset J$ and it is $0$ otherwise.

It is a known result in combinatorics that the columns of the matrix $B$ are
linearly independent. The context is the spectral theory of the Johnson graph,
which is developed in [17, Chapter 6]. To be precise, the desired identity
$\,{\rm rank}(B)=\binom{n}{k-r-1}\,$ can be found in [17, Theorem 6.3.3]. From
this statement we deduce that $\alpha_{I,I}=0$ for all
$I\in\binom{[n]}{k-r-1}$.

For the general case, we fix $I^{\prime}=I_{0}\sqcup K$ and
$J^{\prime}=J_{0}\sqcup K$ where $I_{0},J_{0}\in\binom{[n]}{s}$ are disjoint,
with $s\geq 1$, and $K\subset[n]\backslash(I_{0}\sqcup J_{0})$ has size $k-s$.
From (19) we obtain the equations

$\\!\\!\\!\sum_{\begin{subarray}{c}L\subset K\\\
|L|=r+1\end{subarray}}\\!\\!\epsilon_{(I_{0}\sqcup K)\setminus
L,L}\cdot\epsilon_{(J_{0}\sqcup K)\setminus L,L}\cdot\alpha_{(I_{0}\sqcup
K)\setminus L,(J_{0}\sqcup K)\setminus L}\,=\,0\quad\text{for any
}K\in\binom{[n]\backslash(I_{0}\sqcup J_{0})}{k-s}.$ (20)

Suppressing $I_{0}$ and $J_{0}$ from the indices of $\alpha$, we rewrite (20)
in matrix form $UBVa=0$, where

  * •

$a_{I}=\alpha_{I_{0}\sqcup I,J_{0}\sqcup I}$ for any subset
$I\in\binom{[n]\setminus(I_{0}\sqcup J_{0})}{k-s-r-1}$. To get to (20), we
would set $I=K\backslash L$.

  * •

$B$ is a $\binom{n-2s}{k-s}\times\binom{n-2s}{k-s-r-1}$ matrix with entries in
$\\{0,1\\}$. The columns of $B$ are indexed by subsets $I$ of size
$k{-}s{-}r{-}1$ of $[n]\backslash(I_{0}\sqcup J_{0})$ and the rows are indexed
by subsets $K$ of size $k{-}s$ subsets of $[n]\backslash(I_{0}\sqcup J_{0})$.
The entries are $B_{K,I}=1$ of $I\subset K$ and $0$ otherwise.

  * •

$U$ and $V$ are diagonal matrices of size $\binom{n-2s}{k-s}$ and
$\binom{n-2s}{k-s-r-1}$ respectively, with entries

$U_{K,K}\,=\,(-1)^{N}\quad\text{ and }\quad V_{I,I}\,=\,(-1)^{M},$

where $N$ counts the elements in $I_{0}\sqcup J_{0}$ that are larger than
elements in $K$ and $M$ counts the elements in $I_{0}\sqcup J_{0}$ that are
larger than elements in $I$. In symbols,

$N\,\,=\sum_{i\in I_{0}\sqcup J_{0}}\sum_{\ell\in
K}1_{i>\ell}\qquad\text{and}\qquad M\,\,=\sum_{i\in I_{0}\sqcup
J_{0}}\sum_{\ell\in I}1_{i>\ell}.\vspace{-0.2cm}$

Writing $I\subset K$ and $L=K\backslash I$, we find $\epsilon_{(I_{0}\sqcup
K)\setminus L,L}\cdot\epsilon_{(J_{0}\sqcup K)\setminus L,L}=(-1)^{N+M}$ for
the sign in (20). Again, by virtue of [17, Theorem 6.3.3], we have ${\rm
rank}(B)=\binom{n-2s}{k-s-r-1}$ and hence $a_{I}=\alpha_{I_{0}\sqcup
I,J_{0}\sqcup I}=0$ for any set $I\subset[n]\backslash(I_{0}\sqcup J_{0})$ of
size $k{-}s{-}r{-}1$. Since this holds for any pair of disjoint index sets
$I_{0},J_{0}\in\binom{[n]}{s}$ where $0\leq s\leq k{-}r{-}1$, we deduce that
$\alpha=0$. ∎

We next present a Khovanskii basis [4] for the coordinate ring of the variety
${\rm SH}(k,n,r)$. Khovanskii bases used to be called SAGBI bases in earlier
works, and our arguments follows those given for Grassmannians in [27, Section
3.1] and for flag varieties in [24, Chapter 14].

We fix the reverse lexicographic term order $>$ on the polynomial ring
$S=\mathbb{C}[{\bf x}]$, where $x_{11}>x_{12}>\cdots>x_{n-k+r,n}$. This is a
diagonal term order, i.e. for each minor of ${\bf x}$, the initial monomial is
the product of the entries on its diagonal. Our coordinate ring is the image
of the polynomial ring $R=\mathbb{C}\bigl{[}\langle I\rangle,[J]\,\bigr{]}$
under the ring homomorphism $\phi=\phi_{k,n,r}$ into $S$. See Remark 2.6. For
each of the $2\binom{n}{k}$ generators of $R$, we consider the initial
monomial of its image in $S$. This gives a list of $\binom{n}{k}$ monomials
${\rm in}_{>}\phi(\langle I\rangle)$ of degree $k$ and $\binom{n}{k}$
monomials ${\rm in}_{>}\phi([J])$ of degree $n-k$. These monomials lie in the
initial algebra of our coordinate ring

${\rm in}_{>}(\phi(S))\,\,=\,\,\mathbb{C}\bigl{[}\,{\rm
in}_{>}(f)\,:\,f\in\phi(S)\,\bigr{]}.$ (21)

######  Theorem 3.5.

The $2\binom{n}{k}$ minors $\phi(\langle I\rangle)$ and $\phi([J])$ form a
Khovanskii basis for the coordinate ring $\phi(R)$ of the spinor-helicity
variety ${\rm SH}(k,n,r)$, i.e. their initial monomials generate (21).

###### Proof.

Our argument mirrors that of [24, Theorem 14.11]. We use the set-up in the
proof of Theorem 2.7. Monomials of bidegree $(d_{1},d_{2})$ in $S$ are
represented by skew tableaux. These are formed by placing $d_{1}$ increasing
rows of length $n-k$, shifted by $r$ steps to the right, above $d_{2}$
increasing rows of length $k$. This is seen in (14) for $d_{1}=d_{2}=1$.
Following [24, Lemma 14.13], a monomial in $S$ is the initial monomial of an
element in $\phi(R)$ if and only its representation as a skew tableau, as in
[24, eqn (14.4)], is a semi-standard skew tableau. Hence the initial algebra
(21) is spanned as a $\mathbb{C}$-vector space by ${\bf x}$-monomials that
correspond to semi-standard skew tableaux. Every such monomial is a product of
diagonal monomials ${\rm in}_{>}\phi(\langle I\rangle)$ of degree $k$ and
diagonal monomials ${\rm in}_{>}\phi([J])$ of degree $n-k$. ∎

We now illustrate Theorem 3.5 for the non-trivial instance shown in Figure 2.

######  Example 3.6 ($k=3,n=7,r=1$).

The polynomial ring $R$ is generated by the $35$ entries of

${\bf
x}\,\,=\,\,\small\begin{pmatrix}x_{11}&x_{12}&x_{13}&x_{14}&x_{15}&x_{16}&x_{17}\\\
x_{21}&x_{22}&x_{23}&x_{24}&x_{25}&x_{26}&x_{27}\\\
x_{31}&x_{32}&x_{33}&x_{34}&x_{35}&x_{36}&x_{37}\\\
x_{41}&x_{42}&x_{43}&x_{44}&x_{45}&x_{46}&x_{47}\\\
x_{51}&x_{52}&x_{53}&x_{54}&x_{55}&x_{56}&x_{57}\end{pmatrix}.$ (22)

The polynomial ring $S$ is generated by the $70$ brackets in Figure 2. The map
$\phi:S\rightarrow R$ takes $\langle ijk\rangle$ to the $3\times 3$-minor of
${\bf x}$ with row indices $\\{1,2,3\\}$ and column indices $\\{i,j,k\\}$. It
takes $[ijk]$ to the signed $4\times 4$-minor with row indices $\\{2,3,4,5\\}$
and column indices $[7]\backslash\\{i,j,k\\}$. We consider the image (21) of
the map that takes each bracket to the diagonal initial monomial:

$\begin{matrix}{\rm in}_{>}\phi\,:\,R\rightarrow S:&\\!\\!\\!\\!\langle
123\rangle\,\mapsto\,x_{11}x_{22}x_{33},&\\!\\!\\!\\!\langle
124\rangle\,\mapsto\,x_{11}x_{22}x_{34},&\\!\\!\ldots\,,&\\!\\!\\!\\!\langle
567\rangle\,\mapsto\,x_{15}x_{26}x_{37},\\\ &\,[123]\mapsto
x_{24}x_{35}x_{46}x_{57},&\,[124]\mapsto
x_{23}x_{35}x_{46}x_{57},&\\!\\!\ldots\,,&\,[567]\mapsto
x_{21}x_{32}x_{43}x_{54}.\end{matrix}$

The kernel of the monomial map $\,{\rm in}_{>}\phi\,$ is a toric ideal in $S$.
This is minimally generated by $329$ binomial quadrics. First, there are $140$
quadratic binomials from Young’s poset $Y_{3,7}$:

$\langle 125\rangle\langle 134\rangle-\langle 124\rangle\langle
135\rangle\,,\,\,\langle 126\rangle\langle 134\rangle-\langle
124\rangle\langle 136\rangle\,,\,\,\ldots\,,\,\langle 367\rangle\langle
457\rangle-\langle 357\rangle\langle 467\rangle$ (23)

Likewise, among the generators of the toric ideal $\,{\rm kernel}\bigl{(}{\rm
in}_{>}(\phi)\bigr{)}\,$ are $140$ binomials from $\widetilde{Y}_{3,7}$:

$[125][134]-[124][135]\,,\,\,[126][134]-[124][136]\,,\,\ldots\,,\,[367][457]-[357][467].$
(24)

Third, and most important, there are $49$ mixed binomial quadrics in our toric
ideal:

$\begin{matrix}\langle 123\rangle[123]-\langle 145\rangle[145],&\langle
124\rangle[123]+\langle 145\rangle[135],&\langle 125\rangle[123]-\langle
145\rangle[134],\\\ \qquad\ldots\quad\ldots\quad\ldots\qquad,&\langle
123\rangle[236]+\langle 124\rangle[246],&\langle 123\rangle[237]+\langle
124\rangle[247].\\\ \end{matrix}$ (25)

The initial monomials in (23), (24) and (25) are the incomparable pairs in the
poset $\mathcal{P}_{3,7,1}$. The toric variety defined by our binomials is a
toric degeneration of the spinor-helicity-variety ${\rm SH}(3,7,1)$. The
maximal chains of $\mathcal{P}_{3,7,1}$ form a triangulation of its Newton-
Okounkov body.

We close this section with the remark that the poset $\mathcal{P}_{k,n,r}$ is
a distributive lattice, just like $Y_{k,n}$ and $\widetilde{Y}_{k,n}$. The
join and meet operations $\wedge,\vee$ as follows: if $\langle I\rangle[J]$ is
an incomparable pair, then $\langle I\vee J\rangle$ and $[I\wedge J]$ are
obtained by sorting the columns of the skew Young tableaux (14). With these
lattice operations, the binomials (25) can be written succinctly as follows:

$\langle I\rangle\cdot[J]\,-\,\langle I\vee J\rangle\cdot[I\wedge J].$

In other words, the description in [24, Theorem 14.16] extends to the spinor-
helicity varieties.

##  4 Mandelstam Variety

The componentwise multiplication of two vectors is known as the Hadamard
product. We consider the Hadamard product of two Plücker vectors. This gives
rise to a rational map

$s\,:\,\mathbb{P}^{\binom{n}{k}-1}\times\mathbb{P}^{\binom{n}{k}-1}\,\dashrightarrow\,\mathbb{P}^{\binom{n}{k}-1}.$
(26)

Generalizing the case $k=2$ in (6), the coordinates of $s$ are called
Mandelstam invariants:

$s_{i_{1}i_{2}\cdots i_{k}}\,\,=\,\,\langle i_{1}i_{2}\cdots
i_{k}\rangle[i_{1}i_{2}\cdots i_{k}].$ (27)

We define the Mandelstam variety ${\rm M}(k,n,r)$ to be the closure of the
image of the spinor-helicity variety ${\rm SH}(k,n,r)$ under the Hadamard
product map $s$. Thus, ${\rm M}(k,n,r)$ is an irreducible variety in
$\mathbb{P}^{\binom{n}{k}-1}$. We write $\mathcal{I}({\rm M}(k,n,r))$ for the
homogeneous prime ideal of this variety. This comprises all polynomial
relations among the Mandelstam invariants $\,s_{i_{1}i_{2}\cdots i_{k}}$.

######  Proposition 4.1.

The linear span of the Mandelstam variety ${\rm M}(k,n,r)$ in
$\mathbb{P}^{\binom{n}{k}-1}$ is the subspace $\,\mathbb{P}^{N}$ which is
defined by the momentum conservation relations. Its dimension equals

$N\,\,=\,\,\binom{n}{k}-1-\binom{n}{k-r-1}.$

This refers to the momentum conservation relations in the CEGM model [11, eqn
(5.6)].

###### Proof.

In our notation, the momentum conservation relations are written as follows:

$\sum_{J\in\binom{[n]}{r+1}}\\!\\!s_{IJ}\,\,=\,\,0\qquad\hbox{for
all}\,\,\,I\in\binom{[n]}{k-r-1}.$ (28)

We claim that these linear forms lie in $\mathcal{I}({\rm M}(k,n,r))$ and that
they are linearly independent. To see this, recall the matrix $PQ^{T}$ from
Theorem 3.4. The $\binom{n}{k-r-1}$ diagonal entries of $PQ^{T}$ are
$\,\sum_{J\in\binom{[n]}{r+1}}\langle IJ\rangle[IJ]$, where the index $I$ runs
over $\binom{[n]}{k-r-1}$. This sum agrees with (28), which therefore lies in
$\mathcal{I}({\rm M}(k,n,r))$. The argument with the Johnson matrix in the
proof of Theorem 3.4 shows that our $\binom{n}{k-r-1}$ linear forms are
linearly independent. The dimension count in Corollary 2.10 implies that they
span the space of all linear forms in $\,\mathcal{I}({\rm M}(k,n,r))$. ∎

######  Example 4.2 ($k=3,n=6,r=1$).

There are six momentum conservation relations:

$\begin{matrix}s_{123}+s_{124}+s_{125}+s_{126}+s_{134}+s_{135}+s_{136}+s_{145}+s_{146}+s_{156}&=&0,\\\
s_{123}+s_{124}+s_{125}+s_{126}+s_{234}+s_{235}+s_{236}+s_{245}+s_{246}+s_{256}&=&0,\\\
s_{123}+s_{134}+s_{135}+s_{136}+s_{234}+s_{235}+s_{236}+s_{345}+s_{346}+s_{356}&=&0,\\\
s_{124}+s_{134}+s_{145}+s_{146}+s_{234}+s_{245}+s_{246}+s_{345}+s_{346}+s_{456}&=&0,\\\
s_{125}+s_{135}+s_{145}+s_{156}+s_{235}+s_{245}+s_{256}+s_{345}+s_{356}+s_{456}&=&0,\\\
s_{126}+s_{136}+s_{146}+s_{156}+s_{236}+s_{246}+s_{256}+s_{346}+s_{356}+s_{456}&=&0.\end{matrix}$
(29)

These define a subspace $\mathbb{P}^{13}$ of $\mathbb{P}^{19}$. The variety
${\rm M}(3,6,1)$ has codimension four in this $\mathbb{P}^{13}$. A general
formula for the dimension of any Mandelstam variety is given in the next
result.

######  Proposition 4.3.

The dimension of the Mandelstam variety equals

${\rm dim}({\rm M}(k,n,r))\,\,=\,\,{\rm dim}({\rm
SH}(k,n,r))\,-n+1\,\,=\,\,2k(n-k)-(k-r)^{2}-n+1.$

###### Proof.

The $n$-dimensional torus $(\mathbb{C}^{*})^{n}$ acts on the spinor-helicity
variety as follows:

$\begin{matrix}\langle i_{1}i_{2}\cdots
i_{k}\rangle&\mapsto&t_{i_{1}}\,t_{i_{2}}\,\cdots\,t_{i_{k}}\,\langle
i_{1}i_{2}\cdots i_{k}\rangle,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
[i_{1}i_{2}\cdots i_{k}]&\mapsto&t_{i_{1}}^{-1}t_{i_{2}}^{-1}\cdots
t_{i_{k}}^{-1}[i_{1}i_{2}\cdots i_{k}].\end{matrix}$ (30)

The stabilizer is a one-dimensional torus $\mathbb{C}^{*}$. The ring of
polynomial invariants of the torus action is generated by the Mandelstam
invariants (27). Therefore, ${\rm M}(k,n,r)$ is the image of the quotient map,
and its dimension is $n-1$ less than the dimension of ${\rm SH}(k,n,r)$. ∎

Propositions 4.1 and 4.3 are illustrated in Table 1. For the given values of
$k,n$ and $r$, we display the dimension of ${\rm M}(k,n,r)$ and the dimension
$N$ of its linear span. Note that $N$ can be quite a bit smaller than the
dimension $\binom{n}{k}-1$ of the ambient Plücker space. For example, ${\rm
M}(3,8,0)$ has dimension $14$ inside a linear subspace $\mathbb{P}^{27}$ of
the Plücker space $\mathbb{P}^{55}$.

$\setcounter{MaxMatrixCols}{13}\begin{matrix}&\\!\\!k,n=\\!\\!\\!\\!&2,4&2,5&2,6&2,7&2,8&3,6&3,7&3,8&3,9&4,8&4,9\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
r=0&&1,1&4,4&7,8&10,13&13,19&4,4&9,13&14,27&19,47&9,13&16,41\\\
r=1&&4,4&7,8&10,13&13,19&16,26&9,13&14,27&19,47&24,74&16,41&23,89\\\
r=2&&5,5&8,9&11,14&14,20&17,27&\\!12,18&17,33&22,54&27,82&21,61&\\!28,116\end{matrix}\vspace{-0.3cm}$
Table 1:  The dimension of the Mandelstam variety ${\rm M}(k,n,r)$ and its
ambient space $\mathbb{P}^{N}$.

We next discuss the Mandelstam variety in the case of primary interest in
physics, namely $k=2$. This lives in $\mathbb{P}^{\binom{n}{2}-1}$. Here
$(s_{ij})$ is a symmetric $n\times n$ matrix with $s_{11}=\cdots=s_{nn}=0$.

######  Proposition 4.4.

The $5\times 5$ minors of $(s_{ij})$ vanish on the varieties ${\rm M}(2,n,r)$.
For $r=1$, the sum of all matrix entries is zero. For $r=0$, each row and each
column in $(s_{ij})$ sums to zero, so we only need the
$\frac{1}{2}\bigl{(}\binom{n-1}{5}^{2}+\binom{n-1}{5}\bigr{)}$ minors
involving the last row or last column.

###### Proof.

The sum constraints are the momentum conservations relations in (28). It
suffices to prove the first sentence for $r=2$, when there are no such
relations. Note that ${\rm M}(2,n,2)$ is the Hadamard product [5] of ${\rm
Gr}(2,n)$ with itself. For each matrix $s$ in ${\rm M}(2,n,2)$, we have

$\begin{matrix}s_{ij}\,\,=\,\langle
i\,j\rangle[i\,j]\,=\,(\lambda_{1i}\lambda_{2j}-\lambda_{2i}\lambda_{1j})(\widetilde{\lambda}_{1i}\widetilde{\lambda}_{2j}-\widetilde{\lambda}_{2i}\widetilde{\lambda}_{1j})\qquad\qquad\qquad\qquad\\\
\qquad\qquad\quad=\,\lambda_{1i}\widetilde{\lambda}_{1i}\cdot\lambda_{2j}\widetilde{\lambda}_{2j}\,-\,\lambda_{1i}\widetilde{\lambda}_{2i}\cdot\lambda_{2j}\widetilde{\lambda}_{1j}\,-\,\lambda_{2i}\widetilde{\lambda}_{1i}\cdot\lambda_{1j}\widetilde{\lambda}_{2j}\,+\,\lambda_{2i}\widetilde{\lambda}_{2i}\cdot\lambda_{1j}\widetilde{\lambda}_{1j}.\end{matrix}$
(31)

This shows that $s=(s_{ij})$ is a sum of four matrices of rank one, and hence
${\rm rank}(s)\leq 4$. ∎

Our next theorem states that the equations in Proposition 4.4 generate the
prime ideals.

######  Theorem 4.5.

For $r=0,1,2$, the prime ideal of the Mandelstam variety ${\rm M}(2,n,r)$ is
generated by the $5\times 5$ minors of $(s_{ij})$ together with the respective
linear forms in (28).

###### Proof.

We first record the dimensions of our three Mandelstam varieties from
Proposition 4.3:

${\rm dim}({\rm M}(2,n,2))=3n-7,\,\,{\rm dim}({\rm M}(2,n,1))=3n-8\quad{\rm
and}\quad{\rm dim}({\rm M}(2,n,0))=3n-11.$ (32)

Let $J$ denote the ideal generated by the $5\times 5$ minors. This ideal is
prime. This was shown for $4\times 4$ minors in [14, Theorem 3.4]. The proof
is the same for $5\times 5$ minors. A dimension count shows that $V(J)$ has
dimension $3n-7$, and from this we obtain the first assertion.

Every matrix in ${\rm M}(2,n,2)$ is a product $XX^{T}$ where $X$ is an
$n\times 4$ matrix whose rows lie on the Fermat quadric
$\mathcal{V}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})$. Hence the coordinate
ring of ${\rm M}(2,n,2)$ is the $n$-fold tensor product of the coordinate ring
of the Fermat quadric. The latter ring is a normal domain and hence so is the
former. All associated primes of the principal ideal generated by $\sum
s_{ij}$ have height one in this domain. Using Lemma 4.6 below, we can now
conclude that this principal ideal is a prime ideal. This argument shows that
$J+\langle\,\sum s_{ij}\,\rangle$ is a prime ideal in the polynomial ring
$\mathbb{C}[s]$. The variety of this ideal has codimension $1$ in ${\rm
M}(2,n,2)$. Since this matches the dimension of ${\rm M}(2,n,1)$ in (32), we
conclude that $\mathcal{I}({\rm M}(2,n,1))=J+\langle\,\sum s_{ij}\,\rangle$.

We now turn to $r=0$. Let $K$ be the ideal generated by $J$ and
$\sum_{j=1}^{n}s_{ij}$ for $i=1,2,\ldots,n$. From Proposition 4.4 we know that
${\rm M}(2,n,0)\subseteq\mathcal{V}(K)$. We solve the $n$ linear equations for
$s_{1n},s_{2n},\ldots,s_{n-1,n}$. This leaves us with $\sum_{1\leq i<j\leq
n-1}s_{ij}=0$. Moreover, all $5\times 5$ minors of the $n\times n$ matrix $s$
that involve the index $n$ are sums of $5\times 5$ minors that do not involve
$n$. Using our result for $r=1$, we see that $\mathbb{C}[s]/K$ is isomorphic
to the coordinate ring of ${\rm M}(2,n-1,1)$. This shows that $K$ is prime and
${\rm dim}({\rm V}(K))=3(n-1)-8=3n-11$. This matches the dimension for $r=0$
in (32). We thus conclude that $K$ is the prime ideal of ${\rm M}(2,n,0)$. ∎

To complete the proof of Theorem 4.5, we still need to establish the following
lemma.

######  Lemma 4.6.

The equation $\sum_{1\leq i<j\leq n}s_{ij}=0$ defines a hypersurface that is
is reduced and irreducible in the variety of symmetric $n\times n$ matrices
$(s_{ij})$ with zero diagonal and rank $\leq 4$.

###### Proof.

As in [14, Theorem 3.4], we work in the polynomial ring $\mathbb{C}[X]$. Our
hypersurface is the variety cut out by the ideal $I$, which is generated by
the quadric $\sum_{i,j=1}^{n}\sum_{k=1}^{4}x_{ik}x_{jk}$ together with the $n$
Fermat quadrics $\sum_{k=1}^{4}x_{ik}^{2}$. These quadrics form a regular
sequence in $\mathbb{C}[X]$. Hence $\mathbb{C}[X]/I$ is a complete
intersection ring. By examining the Jacobian matrix of these $n+1$ quadrics,
we can show that this variety is a complete intersection and that its singular
locus has codimension $\geq 2$. Serre’s criterion for normality implies that
the coordinate ring $\mathbb{C}[X]/I$ is normal, so it is a product of normal
domains. This ring being graded, it has no non-trivial idempotents. Hence it
is a normal domain, so $I$ is a prime ideal. ∎

We now turn to the general case $k\geq 3$, with $r$ between $0$ and $k-2$. The
Mandelstam invariants $s_{i_{1}i_{2}\cdots i_{k}}$ form a symmetric
$n\times\cdots\times n$ tensor $s$, where an entry is zero whenever
$\\#\\{i_{1},i_{2},\ldots,i_{k}\\}\leq k-1$. Its two-way marginal is the
symmetric $n\times n$ matrix with entries

$s_{i\,j\,+\cdots+}\,\,\,:=\,\,\,\sum_{l_{3}=1}^{n}\sum_{l_{4}=1}^{n}\cdots\\!\sum_{l_{k}=1}^{n}\\!s_{i\,j\,l_{3}l_{4}\cdots
l_{k}}\quad\hbox{for $\,1\leq i,j\leq n$.}$ (33)

######  Proposition 4.7.

For every tensor $s$ in ${\rm M}(k,n,r)$, the two-way marginal has rank $\leq
4$.

######  Example 4.8 ($k=3,n=6$).

The symmetric $6\times 6\times 6$ tensor $(s_{ijk})$ has only $20$ distinct
nonzero entries $s_{ijk}$ for $1\leq i<j<k\leq 6$. Its two-way marginal is the
$6\times 6$ matrix

$\\!\\!\\!\\!\tiny\begin{bmatrix}0\\!\\!&\\!\\!s_{123}{+}s_{124}{+}s_{125}{+}s_{126}\\!\\!&\\!\\!s_{123}{+}s_{134}{+}s_{135}{+}s_{136}\\!\\!&\\!\\!s_{124}{+}s_{134}{+}s_{145}{+}s_{146}\\!\\!&\\!\\!s_{125}{+}s_{135}{+}s_{145}{+}s_{156}\\!\\!&\\!\\!s_{126}{+}s_{136}{+}s_{146}{+}s_{156}\\\
s_{123}{+}s_{124}{+}s_{125}{+}s_{126}\\!\\!&\\!\\!0\\!\\!&\\!\\!s_{123}{+}s_{234}{+}s_{235}{+}s_{236}\\!\\!&\\!\\!s_{124}{+}s_{234}{+}s_{245}{+}s_{246}\\!\\!&\\!\\!s_{125}{+}s_{235}{+}s_{245}{+}s_{256}\\!\\!&\\!\\!s_{126}{+}s_{236}{+}s_{246}{+}s_{256}\\\
s_{123}{+}s_{134}{+}s_{135}{+}s_{136}\\!\\!&\\!\\!s_{123}{+}s_{234}{+}s_{235}{+}s_{236}\\!\\!&\\!\\!0\\!\\!&\\!\\!s_{134}{+}s_{234}{+}s_{345}{+}s_{346}\\!\\!&\\!\\!s_{135}{+}s_{235}{+}s_{345}{+}s_{356}\\!\\!&\\!\\!s_{136}{+}s_{236}{+}s_{346}{+}s_{356}\\\
s_{124}{+}s_{134}{+}s_{145}{+}s_{146}\\!\\!&\\!\\!s_{124}{+}s_{234}{+}s_{245}{+}s_{246}\\!\\!&\\!\\!s_{134}{+}s_{234}{+}s_{345}{+}s_{346}\\!\\!&\\!\\!0\\!\\!&\\!\\!s_{145}{+}s_{245}{+}s_{345}{+}s_{456}\\!\\!&\\!\\!s_{146}{+}s_{246}{+}s_{346}{+}s_{456}\\\
s_{125}{+}s_{135}{+}s_{145}{+}s_{156}\\!\\!&\\!\\!s_{125}{+}s_{235}{+}s_{245}{+}s_{256}\\!\\!&\\!\\!s_{135}{+}s_{235}{+}s_{345}{+}s_{356}\\!\\!&\\!\\!s_{145}{+}s_{245}{+}s_{345}{+}s_{456}\\!\\!&\\!\\!0\\!\\!&\\!\\!s_{156}{+}s_{256}{+}s_{356}{+}s_{456}\\\
s_{126}{+}s_{136}{+}s_{146}{+}s_{156}\\!\\!&\\!\\!s_{126}{+}s_{236}{+}s_{246}{+}s_{256}\\!\\!&\\!\\!s_{136}{+}s_{236}{+}s_{346}{+}s_{356}\\!\\!&\\!\\!s_{146}{+}s_{246}{+}s_{346}{+}s_{456}\\!\\!&\\!\\!s_{156}{+}s_{256}{+}s_{356}{+}s_{456}\\!\\!&\\!\\!0\end{bmatrix}\\!\\!.$

This matrix has rank four on ${\rm M}(3,6,1)$ and hence also on ${\rm
M}(3,6,0)$, but not on ${\rm M}(3,6,2)$. Geometrically, this matrix encodes
the map ${\rm M}(3,6,1)\dashrightarrow{\rm M}(2,6,0)$ of Mandelstam varieties.

###### Proof of Proposition 4.7.

Since ${\rm M}(n,k,r)\subset{\rm M}(n,k,k-2)$ for all $r<k-2$, it suffices to
prove the statement in the case $r=k-2$. For this case, we consider the
rational map

$\mathrm{SH}(k,n,k-2)\,\dashrightarrow\,\mathrm{SH}(2,n,0)\,,\,\,(V,W)\,\mapsto\,(V\cap
W^{\perp},V^{\perp}\cap W).$ (34)

In terms of the parametrization in Remark 2.6, the space $V^{\perp}\cap W$ is
the kernel of the matrix $X$, while $V\cap W^{\perp}$ is the span of the $k-r$
middle rows, which are indexed by $r+1,\ldots,k$. The Plücker coordinates of
the two spaces can be written as bilinear forms in the given Plücker
coordinates $\langle i_{1}i_{2}\cdots i_{k}\rangle$ and $[i_{1}i_{2}\cdots
i_{k}]$ for $V$ and $W$. The map (34) is equivariant with respect to the
involution in Remark 2.4, and it induces a rational map of Mandelstam
varieties

${\rm M}(k,n,k-2)\,\dashrightarrow\,{\rm M}(2,n,0).$ (35)

In terms of coordinates, the map (35) takes the $n\times\cdots\times n$ tensor
$s$ to its two-way marginal, which is the $n\times n$ matrix with entries
(33). That matrix has rank $\leq 4$, by Proposition 4.4. ∎

######  Remark 4.9.

The rational map in (34) is a well-defined morphism on the open subset
$\mathrm{SH}(k,n,k-2)\backslash\mathrm{SH}(k,n,k-3)$. This open subset is the
smooth locus of $\mathrm{SH}(k,n,k-2)$. In general, the singular locus of the
spinor-helicity variety $\mathrm{SH}(k,n,r)$ is precisely
$\mathrm{SH}(k,n,r-1)$.

The Mandelstam variety ${\rm M}(k,n,r)$ has a natural parametrization, namely
the composition of the Hadamard map with the parametrization given by the map
$\phi_{k,n,r}$ in Remark 2.6. The parameters are the entries in the
$(n-k+r)\times n$ matrix ${\bf x}=(x_{ij})$. The Mandelstam invariant $s_{I}$
is a polynomial in the entries of ${\bf x}$. Namely, $s_{I}$ is the product of
the $k\times k$ minor indexed by $I$ in the first $k$ rows of ${\bf x}$ with
the signed $(n-k)\times(n-k)$ minor indexed by $[n]\backslash I$ in the last
$n-k$ rows of ${\bf x}$. The corresponding ring map $\phi_{k,n,r}\circ s^{*}$
has kernel $\mathcal{I}({\rm M}(k,n,r))$.

We now replace $\phi_{k,n,r}$ with a birational parametrization for (10),
namely by specializing ${\bf x}$ as follows. Start rows $r+1,\ldots,k$ with a
unit matrix. This leaves $(k-r)(n-k+r)$ parameters for $V\cap W^{\perp}$.
Start rows $1,\ldots,r$ with $k-r$ zero columns, followed by a unit matrix.
This leaves $r(n-k)$ parameters for $V/(V\cap W^{\perp})$. Start rows
$k+1,\ldots,n-k+r$ with $k-r$ zero columns, followed by a unit matrix. This
leaves $(n-2k+r)k$ parameters for $W^{\perp}/(V\cap W^{\perp})$.

######  Corollary 4.10.

The rules above give a birational map
$\psi_{k,n,r}:\mathbb{C}^{2k(n-k)-(k-r)^{2}}\rightarrow{\rm SH}(k,n,r)$.

###### Proof.

We first note that the parameter count above matches the dimension formula in
(11):

$(k-r)(n-k+r)\,+\,r(n-k)\,+\,(n-2k+r)k\quad=\quad 2k(n-k)-(k-r)^{2}.$

To parametrize (10), one first chooses $V\cap W^{\perp}$, and thereafter
$V/(V\cap W^{\perp})$ and $W^{\perp}/(V\cap W^{\perp})$. Each block of rows
gives a birational parametrization of the respective Grassmannian. ∎

By composing $\psi_{k,n,r}$ with the Hadamard map $s$, we obtain a
parametrization of the Mandelstam variety ${\rm M}(k,n,r)$. Each fiber has
dimension $n-1$, reflecting the torus action in (30). To obtain a finite-to-
one parametrization of ${\rm M}(k,n,r)$, we can now replace $n-1$ of the
matrix entries $x_{ij}$ by $1$. The resulting parametric representation of
${\rm M}(k,n,r)$ can then be used in numerical algebraic geometry. The
following proposition serves as an illustration.

######  Proposition 4.11.

The Mandelstam variety ${\rm M}(3,6,1)$ has dimension $9$ and degree $56$ in
$\mathbb{P}^{19}$. Its prime ideal is minimally generated by $14$ quartics,
plus the six linear forms in (29). Inside their subspace $\mathbb{P}^{13}$,
the variety is arithmetically Cohen-Macaulay, and its Betti diagram equals

$\,\small\begin{bmatrix}\,14&\cdot&\cdot&\cdot\,\\\
\,\cdot&56&64&21\,\end{bmatrix}.$ (36)

###### Computational proof.

The map $\psi_{3,6,1}$ is given by the following specialization of our matrix:

${\bf x}\,\,=\,\,\begin{bmatrix}\,0&0&1&b_{1}&b_{2}&b_{3}\\\
\,1&0&a_{1}&a_{2}&a_{3}&a_{4}\\\ \,0&1&a_{5}&a_{6}&a_{7}&a_{8}\\\
\,0&0&1&c_{1}&c_{2}&c_{3}\\\ \end{bmatrix}.$

We now remove five parameters by setting $a_{1}=a_{5}=c_{1}=c_{2}=c_{3}=1$.
The Mandelstam invariants are polynomials in the nine remaining unknowns
$a_{i}$ and $b_{j}$. This specifies a two-to-one map
$\,\mathbb{C}^{9}\rightarrow\mathbb{P}^{19}$. A computation checks that the
Jacobian of the map has full rank. Hence the closure of its image is the
$9$-dimensional Mandelstam variety ${\rm M}(3,6,1)$. This lies in the
$\mathbb{P}^{13}$ defined by (29). We use these relations to eliminate the six
variables $\,s_{123},s_{124},s_{134},s_{234},s_{235},s_{236}$. Thereafter, we
can view ${\mathcal{I}}({\rm M}(3,6,1))$ as an ideal in the remaining $14$
variables. Our computations with this ideal were carried out in Macaulay2
[18].

The ideal contains no quadrics or cubics, but we find $14$ linearly
independent quartics. Let $I$ be the subideal generated by the $14$ quartics.
This has codimension $4$ and degree $56$, and it contains the $21$ quintics
given by the $5\times 5$ minors of the $6\times 6$ matrix in Example 4.8. We
compute the minimal free resolution of $I$, and find that its Betti diagram
equals (36). Thus, $I$ is Cohen-Macaulay, and so it is an intersection of
primary ideals of codimension $4$.

We now apply HomotopyContinuation.jl [10] to the map
$\,\mathbb{C}^{9}\rightarrow\mathbb{P}^{19}$, and we compute the degree of its
image. This yields an independent proof that ${\rm M}(3,6,1)$ has degree $56$.
Since $I$ has degree $56$, and since the degree is additive over the primary
components, this shows that the ideal $I$ is prime. We conclude that
$I={\mathcal{I}}({\rm M}(3,6,1))$, and the proof is complete. ∎

######  Remark 4.12.

At present, we do not know the meaning of our $14$ quartic generators. The
shortest quartic we found has $140$ monomials. In reverse lexicographic order,
it looks like

$\small\begin{matrix}s_{136}s_{156}s_{235}s_{345}-s_{135}s_{156}s_{236}s_{345}-s_{145}s_{156}s_{236}s_{345}-s_{156}^{2}s_{236}s_{345}-s_{146}s_{156}s_{245}s_{345}\phantom{moo}\\\
+\,s_{135}s_{156}s_{246}s_{345}+s_{145}s_{156}s_{246}s_{345}+s_{156}^{2}s_{246}s_{345}+s_{136}s_{145}s_{256}s_{345}-s_{135}s_{146}s_{256}s_{345}+\,\cdots\\\
\cdots\,\,\cdots\,\,+\,s_{136}s_{345}s_{456}^{2}+s_{156}s_{345}s_{456}^{2}-s_{135}s_{346}s_{456}^{2}-2s_{135}s_{356}s_{456}^{2}+s_{145}s_{356}s_{456}^{2}-s_{135}s_{456}^{3}.\end{matrix}$

We close this section by recording a few more general facts about Mandelstam
varieties.

######  Proposition 4.13.

The Mandelstam variety ${\rm M}(k,n,k)$ is the Hadamard square of the
Grassmannian ${\rm Gr}(k,n)$. This contains all other Mandelstam varieties by
the chain of inclusions

${\rm M}(k,n,0)\,\subset\,{\rm M}(k,n,1)\,\subset\,{\rm
M}(k,n,2)\,\subset\,\cdots\,\subset\,{\rm M}(k,n,k).$ (37)

There is natural chain of dominant deletion maps, induced by removing columns
in $\lambda$ and $\widetilde{\lambda}$:

${\rm M}(k,n,0)\,\dashrightarrow\,{\rm M}(k,n-1,1)\,\dashrightarrow\,{\rm
M}(k,n-2,2)\,\dashrightarrow\,\,\cdots\,\,\dashrightarrow\,{\rm M}(k,n-k,k).$
(38)

###### Proof and discussion.

The term Hadamard square refers to the Hadamard product of a variety with
itself. For an introduction to Hadamard products of varieties see the book
[5]. We obtain inclusions ${\rm SH}(k,n,r)\subset{\rm SH}(k,n,r+1)$ by
relaxing the rank constraints in (10), and we obtain surjections ${\rm
SH}(k,n,r)\rightarrow{\rm SH}(k,n-1,r+1)$ by deleting the last columns in
$\lambda$ and $\widetilde{\lambda}$ respectively. These maps are compatible
with Hadamard products, so they descend to inclusions ${\rm
M}(k,n,r)\subset{\rm M}(k,n,r+1)$ and surjections ${\rm
M}(k,n,r)\rightarrow{\rm M}(k,n-1,r+1)$. It would be interesting to study the
fibers of the maps in (38). Their dimensions are $0,2,4,\ldots,2k-2$. ∎

##  5 Positivity and Tropicalization

In our last two sections, we set the stage for future research on spinor-
helicity varieties, with a view towards tropical geometry, positive geometry,
and applications to scattering amplitudes.

Bossinger, Drummond and Glew [7] studied the Gröbner fan and positive geometry
of the variety ${\rm SH}(2,5,0)$ in Example 1.1 which they identified with the
Grassmannian ${\rm Gr}(3,6)$. We shall examine this in a broader context. The
following result explains their identification.

######  Proposition 5.1.

For any $k\geq 1$, the varieties $\mathrm{SH}(k,2k{+}1,0)$ and
$\mathrm{SH}(k{+}1,2k{+}1,1)$ are isomorphic and their coordinate ring is
isomorphic to that of the Grassmannian $\mathrm{Gr}(k{+}1,2k{+}2)$.

###### Proof.

The isomorphism between $\mathrm{SH}(k,2k\\!+\\!1,0)$ and
$\mathrm{SH}(k\\!+\\!1,2k\\!+\\!1,1)$ arises because $(V,W)$ is in
$\mathrm{SH}(k\\!+\\!1,2k\\!+\\!1,1)$ if and only if $(V^{\perp},W^{\perp})$
is in $\mathrm{SH}(k,2k\\!+\\!1,0)$. The identification with the Grassmannian
${\rm Gr}(k\\!+\\!1,2k\\!+\\!2)$ uses the specialized parametrization
$\psi_{k,2k+1,0}$ in Corollary 4.10. We introduce a new parameter $z$, to
account for ${\rm dim}({\rm Gr}(k\\!+\\!1,2k\\!+\\!2))=1+{\rm dim}({\rm
SH}(k,2k\\!+\\!1))$, and we augment ${\bf x}$ with one extra column
$(0,0,\ldots,0,z)^{T}$. The new matrix ${\bf x}$ has $k+1$ rows and $2k+2$
columns, and it contains $(k+1)^{2}$ parameters. Its maximal minors give a
birational parametrization of ${\rm Gr}(k\\!+\\!1,2k\\!+\\!2)$ and also of
$\mathrm{SH}(k,2k\\!+\\!1,0)$. The minors involving the extra column are the
Plücker coordinates for $V$. The others are Plücker coordinates for
$W^{\perp}$. ∎

The positive Grassmannian ${\rm Gr}_{+}(k,n)$ is defined by requiring all
Plücker coordinates $\langle i_{1}i_{2}\cdots i_{k}\rangle$ of the subspace
$V$ to be real and positive. We define the dually positive Grassmannian ${\rm
Gr}^{+}(k,n)$ to be ${\rm Gr}_{+}(n-k,n)$ under the identification between $W$
and $W^{\perp}$. Thus ${\rm Gr}^{+}(k,n)$ is an open semialgebraic set
isomorphic to ${\rm Gr}_{+}(n-k,n)$. Its is defined by

${\rm sign}\bigl{(}[j_{1}j_{2}\cdots
j_{k}]\bigr{)}\,=\,(-1)^{j_{1}+j_{2}+\cdots+j_{k}}\quad\hbox{for}\quad 1\leq
j_{1}<j_{2}<\cdots<j_{k}\leq n.$ (39)

The positive spinor-helicity variety consists of all positive points in the
spinor-helicity variety:

${\rm SH}_{+}(k,n,r)\,\,\,:=\,\,\,{\rm SH}(k,n,r)\,\,\cap\,\,\bigl{(}\,{\rm
Gr}_{+}(k,n)\,\times\,{\rm
Gr}^{+}(k,n)\bigr{)}\quad\subset\,\,\,\mathbb{R}\mathbb{P}^{\binom{n}{k}-1}\times\mathbb{R}\mathbb{P}^{\binom{n}{k}-1}.$
(40)

We finally define the positive Mandelstam variety $\,{\rm M}_{+}(k,n,r)$ by
the inequalities seen in (39):

${\rm sign}\bigl{(}s_{j_{1}j_{2}\cdots
j_{k}}\bigr{)}\,=\,(-1)^{j_{1}+j_{2}+\cdots+j_{k}}\quad\hbox{for}\quad 1\leq
j_{1}<j_{2}<\cdots<j_{k}\leq n.$ (41)

Thus ${\rm M}_{+}(k,n,r)$ is a semialgebraic subset of
$\mathbb{P}^{\binom{n}{k}-1}$. It contains the image of ${\rm SH}_{+}(k,n,r)$
under the Hadamard product map $s$ in (26). In general, this inclusion is
strict. For example, ${\rm M}_{+}(2,4.2)$ strictly contains the Hadamard
product of $\mathrm{Gr}_{+}(2,4)$ and $\mathrm{Gr}^{+}(2,4)$. To see this, we
note that the following expression is positive on the latter set but not on
the former set:

$s_{13}s_{24}+s_{14}s_{23}-s_{12}s_{34}\,\,=\,\,\langle 13\rangle\langle
24\rangle[14][23]\,+\,\langle 14\rangle\langle 23\rangle[13][24].$ (42)

We now recycle Example 1.1 and Proposition 4.4 for our running example in this
section.

######  Example 5.2 ($k=2,n=5$).

For points in ${\rm SH}_{+}(2,5,r)$, the rank $2$ matrices $P$ and $Q$ satisfy

${\rm sign}(P)\,\,=\,\,\footnotesize\begin{bmatrix}0&+&+&+&+\\\ -&0&+&+&+\\\
-&-&0&+&+\\\ -&-&-&0&+\\\ -&-&-&-&0\end{bmatrix}\qquad{\rm and}\qquad{\rm
sign}(Q)\,\,=\,\,\footnotesize\begin{bmatrix}0&-&+&-&+\\\ +&0&-&+&-\\\
-&+&0&-&+\\\ +&-&+&0&-\\\ -&+&-&+&0\end{bmatrix}.$

The positive spinor-helicity variety ${\rm SH}_{+}(2,5,0)$ is the subset
defined by the equations in (2).

The positive Mandelstam variety ${\rm M}_{+}(2,5,2)$ is a $9$-dimensional
simplex $\mathbb{R}\mathbb{P}_{+}^{9}$. It consists of symmetric $5\times 5$
matrices $s=(s_{ij})$ with alternating sign pattern, i.e. ${\rm sign}(s)={\rm
sign}(Q)$. Its subset ${\rm M}_{+}(2,5,0)$ is defined in this simplex by the
equations $\sum_{j=1}^{5}s_{ij}=0$ for $i=1,2,3,4,5$. This is a cyclic
$4$-polytope with $6$ vertices. It has the f-vector $(6,15,18,9)$. The
vertices are:

$\tiny\begin{pmatrix}0&-&+&0&0\\\ -&0&0&+&0\\\ +&0&0&-&0\\\ 0&+&-&0&0\\\
0&0&0&0&0\end{pmatrix},\begin{pmatrix}0&0&+&-&0\\\ 0&0&-&+&0\\\ +&-&0&0&0\\\
-&+&-&0&0\\\ 0&0&0&0&0\end{pmatrix},\begin{pmatrix}0&-&0&0&+\\\ -&0&0&+&0\\\
0&0&0&0&0\\\ 0&+&0&0&-\\\ +&0&0&-&0\end{pmatrix},\begin{pmatrix}0&0&0&-&+\\\
0&0&0&+&-\\\ 0&0&0&0&0\\\ -&+&0&0&0\\\
+&-&0&0&0\end{pmatrix},\begin{pmatrix}0&0&0&0&0\\\ 0&0&-&+&0\\\ 0&-&0&0&+\\\
0&+&0&0&-\\\ 0&0&+&-&0\end{pmatrix},\begin{pmatrix}0&-&+&0&0\\\ -&0&0&+&0\\\
+&0&0&-&0\\\ 0&+&-&0&0\\\ 0&0&0&0&0\end{pmatrix}.$

Nine of the inequalities ${\rm sign}(s_{ij})=(-1)^{i+j}$ define facets. Only
$s_{24}\geq 0$ is not facet-defining.

The second thread in this section is tropical geometry. Using notation from
the textbook [22], the tropicalizations of our varieties ${\rm SH}(k,n,r)$ and
${\rm M}(k,n,r)$ are the tropical varieties

${\rm trop}({\rm
SH}(k,n,r))\,\,\subset\,\,\mathbb{R}^{\binom{n}{k}}\\!/\mathbb{R}^{\binom{n}{k}\bf
1}\,\times\,\mathbb{R}^{\binom{n}{k}}\\!/\mathbb{R}{\bf 1}\,\,\quad{\rm
and}\quad\,\,{\rm trop}({\rm
M}(k,n,r))\,\,\subset\,\,\mathbb{R}^{\binom{n}{k}}\\!/\mathbb{R}{\bf 1}.$ (43)

These are balanced polyhedral fans whose dimensions are given by Propositions
2.5 and 4.3. Each such fan is a finite intersection of the tropical
hypersurfaces given by a tropical basis. We illustrate these concepts for an
example where the underlying variety is a linear space.

######  Example 5.3 ($k=2,n=5$).

The tropical linear space ${\rm trop}({\rm M}(2,5,0))$ is a pointed fan of
dimension $4$ in $\mathbb{R}^{10}/\mathbb{R}{\bf 1}$. A minimal tropical basis
consists of $15$ linear forms with four terms:

$\begin{matrix}s_{12}{+}s_{13}{+}s_{14}{+}s_{15},s_{12}{+}s_{23}{+}s_{24}{+}s_{25},s_{13}{+}s_{23}{+}s_{34}{+}s_{35},s_{14}{+}s_{24}{+}s_{34}{+}s_{45},s_{15}{+}s_{25}{+}s_{35}{+}s_{45},\\\
s_{12}{+}s_{13}{+}s_{23}{-}s_{45},s_{12}{+}s_{14}{+}s_{24}{-}s_{35},s_{12}{+}s_{15}{+}s_{25}{-}s_{34},s_{34}{+}s_{35}{+}s_{45}{-}s_{12},s_{13}{+}s_{14}{+}s_{34}{-}s_{25},\\\
s_{13}{+}s_{15}{+}s_{35}{-}s_{24},s_{24}{+}s_{25}{+}s_{45}{-}s_{13},s_{14}{+}s_{15}{+}s_{45}{-}s_{23},s_{23}{+}s_{25}{+}s_{35}{-}s_{14},s_{23}{+}s_{24}{+}s_{34}{-}s_{15}.\end{matrix}$

The underlying rank $5$ matroid on $10$ elements is the exceptional unimodular
matroid $R_{10}$. This matroid is seen in [29, Section 3.3]. Hence ${\rm
trop}({\rm M}(2,5,0))$ is the cone over the Bergman complex of $R_{10}$, which
consists of $315$ tetrahedra and $45$ bipyramids. Its f-vector is
$(40,240,510,360)$. The $40$ vertices are the $10$ coordinate points $e_{ij}$
and the $30$ circuits of $R_{10}$.

Combining our two threads, and using the notion of positivity defined above,
leads us to

${\rm trop}_{+}({\rm
SH}(k,n,r))\,\,\subset\,\,\mathbb{R}^{\binom{n}{k}}\\!/\mathbb{R}{\bf
1}\,\times\,\mathbb{R}^{\binom{n}{k}}\\!/\mathbb{R}{\bf 1}\,\,\quad{\rm
and}\quad\,\,{\rm trop}_{+}({\rm
M}(k,n,r))\,\,\subset\,\,\mathbb{R}^{\binom{n}{k}}\\!/\mathbb{R}{\bf 1}.$ (44)

These positive tropical varieties are subfans of the respective tropical
varieties. They are defined by requiring sign compatibility in the tropical
equations. For details see [2, 8, 26].

######  Example 5.4 ($k=2,n=5$).

Following Examples 5.2 and 5.3, the positive tropical Mandelstam variety ${\rm
trop}_{+}({\rm M}(2,5,0))$ is defined by the following system of $15$ tropical
equations:

$\begin{matrix}s_{13}\oplus s_{15}=s_{12}\oplus
s_{14}\,,\,\,s_{24}=s_{12}\oplus s_{23}\oplus s_{25}\,,\,\,s_{13}\oplus
s_{35}=s_{23}\oplus s_{34}\,,\,\,s_{24}=s_{14}\oplus s_{34}\oplus s_{45},\\\
s_{15}\oplus s_{35}=s_{25}\oplus s_{45}\,,\,\,s_{13}\oplus s_{45}=s_{12}\oplus
s_{23}\,,\,\,s_{24}=s_{12}\oplus s_{14}\oplus s_{35}\,,\,\,s_{15}\oplus
s_{34}=s_{12}\oplus s_{25},\\\ s_{12}\oplus s_{35}=s_{34}\oplus
s_{45}\,,\,\,s_{13}\oplus s_{25}=s_{14}\oplus s_{34}\,,\,\,s_{13}\oplus
s_{15}\oplus s_{35}=s_{24}\,,\,\,s_{24}=s_{13}\oplus s_{25}\oplus s_{45},\\\
s_{15}\oplus s_{23}\,=\,s_{14}\oplus s_{45}\,,\,\,\,s_{14}\oplus
s_{35}\,=\,s_{23}\oplus s_{25}\,,\,\,\,s_{24}\,=\,s_{15}\oplus s_{23}\oplus
s_{34},\end{matrix}$

Here $\,x\oplus y:={\rm min}(x,y)$. Note the special role of the non-facet
variable $s_{24}$. These $15$ equations define the positive Bergman complex,
in the sense of Ardila, Klivans and Williams [2].

We find that $\,{\rm trop}_{+}({\rm M}(2,5,0))$ is the cone over a $3$-sphere.
That sphere is glued from $48$ tetrahedra and $18$ bipyramids, and its
f-vector is $(24,108,150,66)$. Geometrically, this is a subdivision of the
boundary of the $4$-polytope $\Delta_{2}\times\Delta_{2}$, which is the
product of two triangles. This is dual to the cyclic polytope in Example 5.2,
and its f-vector is $(9,18,15,6)$. The “fine subdivision” discussed in [2,
Corollary 3.5] is the barycentric subdivision of ${\rm
Bdr}(\Delta_{2}\times\Delta_{2})$.

The positive and tropical geometry of Grassmannians and flag varieties have
been studied intensely in recent years. See [6, 9, 25, 26] for some references
and [11, 15, 21] for physics perspectives. Results from this literature apply
directly to some spinor-helicity varieties. We know from Proposition 2.5 that
$\,{\rm SH}(k,n,0)={\rm Fl}(k,n-k;\mathbb{C}^{n})$ and ${\rm SH}(k,n,k)={\rm
Gr}(k,n)\times{\rm Gr}(k,n)$. The positive geometry structure on these
varieties is established in [20, Section 3.4].

######  Corollary 5.5.

The positive spinor-helicity variety ${\rm SH}_{+}(k,n,r)$ is a positive
geometry for $r=0$ and $r=k$, with boundary structure and canonical form
induced by known results on flag varieties and Grassmannians. The analogous
statement holds for (positive) tropicalizations.

######  Remark 5.6.

A detailed study of ${\rm trop}_{+}({\rm SH}(1,n,0))$ was carried out by
Olarte in [25]. It would desirable to extend this to $k\geq 2$ using the
techniques that were introduced in [6, 9].

Another situation where results can be transferred directly is that seen in
Proposition 5.1.

######  Corollary 5.7.

Modulo a scaling action by $\mathbb{R}^{+}$, the positive Grassmannian
$\mathrm{Gr}(k{+}1,2k{+}2)$ coincides with the positive spinor-helicity
varieties $\mathrm{SH}_{+}(k,2k{+}1,0)$ and $\mathrm{SH}_{+}(k{+}1,2k{+}1,1)$,
for all $k\geq 1$. The analogous statement holds for their (positive) tropical
varieties.

######  Remark 5.8.

We obtain detailed textbook descriptions of ${\rm trop}(\mathrm{SH}(2,5,0))$
from those for ${\rm trop}({\rm Gr}(3,6))$ in [22, Sections 4.4 and 5.4]. This
was pointed out in [7, Section 6]. Similarly, ${\rm
trop}_{+}(\mathrm{SH}(3,7,0))$ arises from ${\rm trop}_{+}({\rm Gr}(4,8))$.
The latter fan was studied in [15, Section 2].

We finally turn to the tropical Mandelstam variety. Recall that ${\rm
M}(k,n,r)$ is the image of ${\rm SH}(k,n,r)$ under the Hadamard product map
$s$ in (26). The tropicalization of this map,

${\rm trop}(s)\,\,:\,\,\mathbb{R}^{\binom{n}{k}}/\mathbb{R}{\bf
1}\,\times\,\mathbb{R}^{\binom{n}{k}}/\mathbb{R}{\bf
1}\,\,\rightarrow\,\,\mathbb{R}^{\binom{n}{k}}/\mathbb{R}{\bf 1},$ (45)

computes the sum of two tropical Plücker vectors, modulo global tropical
scaling. It follows from [22, Theorem 5.5.1] that the Hadamard product map $s$
commutes with tropicalization. The following tropical constructions are thus
obtained directly from their classical analogues.

######  Corollary 5.9.

The tropical Mandelstam variety is the image of the tropical spinor-helicity
variety under the sum map (45). In symbols, for all values of the parameters
$k,n,r$, we have

${\rm trop}({\rm M}(k,n,r))\,\,=\,\,{\rm trop}(s)\bigl{(}\,{\rm trop}({\rm
SH}(k,n,r))\,\bigr{)}.$ (46)

For the parameter values in Corollaries 5.5 and 5.7, the tropical Mandelstam
variety is the image of a tropical Grassmannian or a tropical flag variety
under the sum map. The largest tropical Mandelstam variety ${\rm M}(k,n,k)$ is
the Minkowski sum of the Grassmannian with itself:

${\rm trop}({\rm M}(k,n,k))\,\,=\,\,{\rm trop}({\rm Gr}(k,n))\,+\,{\rm
trop}({\rm Gr}(k,n)).$ (47)

######  Example 5.10 ($k=2,n=5$).

Equation (46) describes a $2$-to-$1$ map from ${\rm trop}({\rm Gr}(3,6))$ onto
${\rm trop}({\rm M}(2,5,0))$. Starting from the census in [22, Example
4.4.10], we can examine this map on every cone of ${\rm trop}({\rm Gr}(3,6))$.
This fan has $65=20+15+30$ rays, grouped into types E, F and G. These rays map
to the $40=10+15+15$ rays of ${\rm trop}({\rm M}(2,5,0))={\rm trop}(R_{10})$.

##  6 The Scattering Correspondence

Scattering amplitudes in the CEGM model [11] are derived from the scattering
potential

$L_{s}\,\,=\,\,\sum_{I\in\binom{[n]}{k}}s_{I}\cdot{\rm log}(p_{I}).$

We assume that $s=(s_{I})$ is a fixed point in the Mandelstam variety ${\rm
M}(k,n,r)$ where $r\leq k-2$. The unknowns $p=(p_{I})$ are the Plücker
coordinates of the open Grassmannian ${\rm Gr}(k,n)^{o}$, which is defined by
$p_{I}\not=0$ for all $I\in\binom{[n]}{k}$. The momentum conservation
relations on ${\rm M}(k,n,k-2)$ ensure that the scattering potential is a
well-defined on the quotient space

$X(k,n)\,\,=\,\,{\rm Gr}(k,n)/(\mathbb{C}^{*})^{n}.$ (48)

This is a very affine variety of dimension $(n-k-1)(k-1)$; see [1, 21]. It is
the moduli space of configurations of $n$ labeled points in linearly general
position in the projective space $\mathbb{P}^{k-1}$.

The scattering potential $L_{s}$ serves as log-likelihood function in
algebraic statistics [28]. In both statistics and physics, one cares about the
critical points of $L_{s}$. These are defined by

$\nabla_{p}\,L_{s}\,\,=\,\,0.$ (49)

We now let both $s$ and $p$ vary, and we consider all solution pairs $(s,p)$
to the system of equations in (49). The pairs $(s,p)$ satisfying (49) are the
points of the scattering correspondence

${\rm C}(k,n,r)\,\,\subset\,\,{\rm M}(k,n,r)\,\times\,{\rm X}(k,n).$

The aim of this section is to initiate the mathematical study of this variety.
We also consider the lifted scattering correspondence, where ${\rm M}(k,n,r)$
is replaced by the spinor-helicity variety:

$\widetilde{\rm C}(k,n,r)\,\,\subset\,\,{\rm SH}(k,n,r)\,\times\,{\rm
X}(k,n).$

######  Example 6.1 ($k=3,n=6$).

We represent six points in $\mathbb{P}^{2}$ by the columns of a matrix

$P\quad=\quad\begin{bmatrix}1&0&0&1&1&1\\\ 0&1&0&1&x&y\\\
0&0&1&1&z&w\end{bmatrix}$ (50)

Hence $x,y,z,w$ are coordinates on the moduli space ${\rm X}(3,6)$. The
scattering potential equals

$\footnotesize\begin{matrix}L_{s}&\\!\\!=\\!\\!&\,\,\,s_{125}\cdot{\rm
log}(z)+s_{126}\cdot{\rm log}(w)+s_{135}\cdot{\rm log}(-x)+s_{136}\cdot{\rm
log}(-y)+s_{145}\cdot{\rm log}(z-x)+s_{146}\cdot{\rm log}(w-y)\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\ &&\\!\\!\\!\\!\\!+\,\,s_{156}\cdot{\rm log}(wx-
yz)+s_{245}\cdot{\rm log}(1-z)+s_{246}\cdot{\rm log}(1-w)+s_{256}\cdot{\rm
log}(z-w)+s_{345}\cdot{\rm log}(x-1)\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&&+\,\,s_{346}\cdot{\rm log}(y-1)\,+\,s_{356}\cdot{\rm
log}(y-x)\,+\,s_{456}\cdot{\rm log}(wx-yz-w-x+y+z).\end{matrix}$

The scattering equations are given by the partial derivatives of the
scattering potential $L_{s}$:

$\begin{matrix}s_{126}\,\frac{1}{w}+s_{146}\,\frac{1}{w-y}+s_{156}\,\frac{x}{wx-
yz}-s_{246}\,\frac{1}{1-w}-s_{256}\,\frac{1}{z-w}+s_{456}\,\frac{x-1}{wx-yz-
w-x+y+z}&=&0,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
s_{135}\,\frac{1}{x}-s_{145}\,\frac{1}{z-x}+s_{156}\,\frac{w}{wx-
yz}+s_{345}\,\frac{1}{x-1}-s_{356}\,\frac{1}{y-x}+s_{456}\,\frac{w-1}{wx-yz-
w-x+y+z}&=&0,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
s_{136}\,\frac{1}{y}-s_{146}\,\frac{1}{w-y}-s_{156}\,\frac{z}{wx-
yz}+s_{346}\,\frac{1}{y-1}+s_{356}\frac{1}{y-x}+s_{456}\,\frac{1-z}{wx-yz-
w-x+y+z}&=&0,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\,s_{125}\,\frac{1}{z}+s_{145}\,\frac{1}{z-x}\,-\,s_{156}\,\frac{y}{wx-
yz}-s_{245}\,\frac{1}{1-z}+s_{256}\,\frac{1}{z-w}+s_{456}\,\frac{1-y}{wx-yz-
w-x+y+z}&=&0.\end{matrix}$ (51)

If the $s_{ijk}$ are general solutions to the linear equations in (29) then
(51) has $26$ complex solutions in ${\rm X}(3,6)$. We are interested in the
case when $s$ lies in ${\rm M}(3,6,1)$, or when we lift to ${\rm SH}(3,6,1)$
by substituting $s_{ijk}=\langle ijk\rangle[ijk]$. We obtain $26$-to-$1$ maps
from the two scattering correspondences ${\rm C}(3,6,1)$ or $\widetilde{\rm
C}(3,6,1)$ onto their kinematic spaces ${\rm M}(3,6,1)$ or ${\rm SH}(3,6,1)$.

We next examine the case of primary interest in physics, namely $k=2$. The
Mandelstam variety ${\rm M}(2,n,r)$ was characterized in Section 4. Here,
$X(2,n)$ is the moduli space $\mathcal{M}_{0,n}$ of $n$ distinct labeled
points $x_{1},x_{2},\ldots,x_{n}$ in
$\mathbb{P}^{1}=\mathbb{C}\cup\\{\infty\\}$. The scattering potential equals

$L_{s}\,\,\,=\,\sum_{1\leq i<j\leq n}\\!\\!s_{ij}\cdot{\rm log}(x_{i}-x_{j}).$

The system of scattering equations $\nabla_{x}L_{s}=0$ can be written
explicitly as follows:

$\sum_{j=1}^{n}\frac{s_{ij}}{x_{i}-x_{j}}\,\,=\,\,0\qquad{\rm
for}\,\,i=1,2,\ldots,n.$ (52)

Let $z$ be a new unknown and consider the following rational function in $z$
of degree $-2$:

$T(z)\,\,\,=\,\sum_{1\leq i<j\leq n}\frac{s_{ij}}{(z-x_{i})(z-x_{j})}.$ (53)

######  Proposition 6.2.

The rational function $T(z)$ is identically zero if and only if (52) holds.

###### Proof.

The residue of $T(z)$ at $z=x_{i}$ is precisely the left hand side of the
equation in (52). The residues at the $n$ poles are all zero if and only if
$T(z)$ is the zero function. ∎

This following result is known in the particle physics literature due to work
of Witten, Roiban-Spradlin-Volovich and Cachazo-He-Yuan [13]. We learned it
from the recent lectures by Thomas Lam [21, Section 4.4]. See the discussion
of “sectors” in [21, Introduction].

######  Theorem 6.3.

The lifted scattering correspondence $\widetilde{\rm C}(2,n,0)$ has $n-3$
irreducible components $\widetilde{\rm C}_{2},\widetilde{\rm
C}_{3},\ldots,\widetilde{\rm C}_{n-2}$. Each of them has the same dimension as
${\rm SH}(2,n,0)$. The irreducible component $\widetilde{\rm C}_{\ell}$
parametrizes all $3$-step flags $V\subseteq U\subseteq W^{\perp}$ where
$(V,W)$ is a point in ${\rm SH}(2,n,0)$ and $U$ is the row span of an
$\ell\times n$ Vandermonde matrix
$(x_{j}^{i})_{i=0,\ldots,\ell-1;j=1,\ldots,n}$. The map from $\widetilde{\rm
C}_{\ell}$ to ${\rm SH}(2,n,0)$ is finite-to-one: its degree is the Eulerian
number $A(n-3,\ell-2)$.

######  Remark 6.4.

The maximum likelihood degree of the moduli space $\mathcal{M}_{0,n}$ equals
$(n-3)!$. In other words, the equations (52) have $(n-3)!$ solutions, provided
$\sum_{j=1}^{n}s_{ij}=0$ for all $i$. See e.g. [28, Proposition 1]. Theorem
6.3 is a geometric realization of the combinatorial identity

$(n-3)!\,\,=\,\,A(n-3,0)+A(n-3,1)+\,\cdots\,+A(n-3,n-4).$ (54)

Note that the Eulerian numbers can be defined by $A(2,0)=A(2,1)=1$ and the
recursions

$A(n-3,\ell-2)\,\,=\,\,(\ell-1)\cdot A(n-4,\ell-2)\,+\,(n-\ell-1)\cdot
A(n-4,\ell-3)\quad{\rm for}\,\,n\geq 3.$ (55)

###### Proof of Theorem 6.3.

A proof for the first part of the statement concerning the irreducible
components of $\widetilde{\rm C}(2,n,0)$ was given in [21, Proposition 4.6 and
Section 4.5]. The argument for the second part was outlined in [13, Section
1.1]. It is based on the degeneration technique known in physics as soft
limits. We now present details from the algebraic perspective of [1].

The variety $\widetilde{\rm C}_{\ell}$ is the image of the following map into
the lifted scattering correspondence:

$(\mathbb{C}[z]_{\leq\ell-1})^{2}\times(\mathbb{C}[z]_{\leq
n-\ell-1})^{2}\times\mathcal{M}_{0,n}\,\rightarrow\,\widetilde{\rm
C}(2,n,0)\,,\,\,(\tau,\widetilde{\tau},x)\,\mapsto\,(\lambda,\widetilde{\lambda},x).$
(56)

Here $x=(x_{1},\ldots,x_{n})\in\mathbb{C}^{n}$ represents a point in
$\mathcal{M}_{0,n}$, the column vector $\tau$ resp. $\widetilde{\tau}$
consists of two polynomials in one variable $z$ of degrees $\ell-1$ resp.
$n-\ell-1$, the $i$th column of $\lambda$ equals $\,\tau(x_{i})\,$, and the
$i$th column of $\,\widetilde{\lambda}\,$ is
$\,\widetilde{\tau}(x_{i})\prod_{j\not=i}(x_{i}-x_{j})^{-1}$. Note that the
row spaces $V$ and $W$ of these $2\times n$ matrices satisfy $V\subseteq
U\subseteq W^{\perp}$, where $U$ is the row space of $(x^{i}_{j})$.

Write $D(n,\ell)$ for the degree of the finite-to-one map $\,\widetilde{\rm
C}_{\ell}\rightarrow{\rm
SH}(2,n,0),\,(\lambda,\widetilde{\lambda},x)\mapsto(\lambda,\widetilde{\lambda})$.
In words, $D(n,\ell)$ is the number of solutions to the scattering equations
in the $\ell$th component of $\widetilde{\rm C}(2,n,0)$. Direct computations
reveal $D(4,2)=D(5,2)=D(5,3)=1$. We claim that

$D(n,\ell)\,\,=\,\,(\ell-1)\cdot D(n-1,\ell)\,+\,(n-\ell-1)\cdot
D(n-1,\ell-1)\quad{\rm for}\,\,n\geq 3.$ (57)

This claim implies Theorem 6.3, because the recursion in (57) matches the
recursion in (55).

It remains to prove (57). This is done using the technique of soft limits. We
drive the $n$th particle to zero, by replacing the Mandelstam coordinate
$s_{ij}$ with $\epsilon\cdot s_{ij}=\epsilon\cdot\langle ij\rangle[ij]$. This
degeneration is compatible with the parametrization (56). As $\epsilon$ tends
to zero, either $\tau(x_{n})=0$ or $\widetilde{\tau}(x_{n})=0$ holds in the
limit. This yields one equation in one unknowns $x_{n}$ of degree $\ell-1$
resp. $n-\ell-1$. The remaining unknowns $x_{1},\ldots,x_{n-1}$ satisfy the
scattering equations on the components of $\widetilde{\rm C}(2,n-1,0)$ that
are indexed by $\ell$ and $\ell-1$ respectively. So, for $\epsilon$ near $0$,
the size of the fibers of $\,\widetilde{\rm C}_{\ell}\rightarrow{\rm
SH}(2,n,0)\,$ is the right hand side of (57). ∎

The irreducible components $\widetilde{\rm C}_{i}$ are referred to as
“sectors” in the physics literature. Section 5.1 in [11] starts with the
sentence “In the k = 2 case it is well known that solutions of the scattering
equations split into $n-3$ sectors”. Our proof was written for mathematicians.

Theorem 6.3 was stated for the lifted scattering correspondence
$\widetilde{\rm C}(2,n,0)$. From our perspective, it is more natural to focus
on the scattering correspondence ${\rm C}(2,n,0)$ because the parameters in
$L_{s}$ are the Mandelstam invariants. The scattering correspondence ${\rm
C}(2,n,0)$ lives over the Mandelstam variety ${\rm M}(2,n,0)$, whose prime
ideal we presented in Theorem 4.5.

######  Corollary 6.5.

The scattering correspondence ${\rm C}(2,n,0)$ has $\lceil\frac{n-3}{2}\rceil$
irreducible components. The varieties ${\rm C}_{\ell}$ and ${\rm C}_{n-\ell}$
in Theorem 6.3 are identified by the map ${\rm SH}(2,n,0)\rightarrow{\rm
M}(2,n,0)$.

###### Proof.

The Hadamard product map $s$ gives rise to a map from the lifted scattering
correspondence in ${\rm SH}(2,n,0)\times\mathcal{M}_{0,n}$ onto the scattering
correspondence in ${\rm M}(2,n,0)\times\mathcal{M}_{0,n}$. This map is a
covering of degree two. The fibers are comprised of solutions for $\ell$ and
for $n-\ell$. These are distinct, unless $\ell=n/2$, where the solution is
fixed under this involution. ∎

We now turn to the case $k=3,r=1$, which was discussed by Cachazo et al. in
[11, Section 5.1]. We revisit their results from an algebraic perspective, and
we report on the identification of irreducible components with the help of
HomotopyContinuation.jl [10].

The moduli space $X(3,n)$ is a quotient of the Grassmannian ${\rm Gr}(3,n)$,
by (48). Hence there are two tautological maps from ${\rm SH}(3,n,1)$ to
$X(3,n)$. These rational maps take the pair $(V,W)$ in (10) to the images of
$V$ and $W$ in $X(3,n)$ respectively. Furthermore, we consider the Veronese
map $v$ from $\mathcal{M}_{0,n}$ into $X(3,n)$ which take $n$ points in
$\mathbb{P}^{1}$ to $n$ points on a conic in $\mathbb{P}^{2}$. Algebraically,
the map $v$ takes a $2\times n$ matrix with $i$th column $(u_{i},v_{i})$ to
the $3\times n$ matrix with $i$th column $(u_{i}^{2},u_{i}v_{i},v_{i}^{2})$.
Its Plücker coordinates satisfy $\langle ijk\rangle\,=\,\langle
ij\rangle\langle ik\rangle\langle jk\rangle$.

We now compose the rational map in (34) with the two tautological maps for
$k=2$, followed by the Veronese map $v$. In this manner, we obtain two
tautological Veronese maps

${\rm SH}(3,n,1)\,\,\dashrightarrow\,\,{\rm
SH}(2,n,0)\,\,\dashrightarrow\,\,X(2,n)\,=\,\mathcal{M}_{0,n}\,\,\stackrel{{\scriptstyle
v}}{{\hookrightarrow}}\,\,X(3,n).$

Explicitly, these two rational maps are $\,(V,W)\mapsto v(V\cap W^{\perp})\,$
and $\,(V,W)\mapsto v(V^{\perp}\cap W)$. We paraphrase the construction in
[11, Section 5.1] in terms of the scattering correspondence.

######  Theorem 6.6.

The lifted scattering correspondence $\widetilde{\rm C}(3,n,1)$ contains four
irreducible components which map birationally onto the moduli space $X(3,n)$.
These are given by the two tautological maps and the two tautological Veronese
maps. In other words, for general kinematic data $(V,W)\in\mathrm{SH}(3,n,1)$,
with Mandelstam invariants $s\in{\rm M}(3,n,1)$, the configurations
$V,\,W,\,v(V\cap W^{\perp}),\,v(V^{\perp}\cap W)$ in $X(3,n)$ are solutions to
the scattering equations (49).

We illustrate this result by solving the equations (51) for a random numerical
instance.

######  Example 6.7 ($k=3,n=6$).

Let $(V,W)\in{\rm SH}(3,6,1)$ be the point with parameter matrix

${\bf x}\,\,\,=\,\,\,\small\begin{bmatrix}4&0&7&4&9&1\\\ 1&3&7&2&8&9\\\
1&7&9&8&0&5\\\ 6&6&2&2&4&2\end{bmatrix}.$

Thus $V$ is the span of the first three rows and $W$ is the kernel of the last
three rows. The Mandelstam invariants are computed by composing the map
$\phi_{3,6,1}$ with the Hadamard map:

$\small\setcounter{MaxMatrixCols}{11}\begin{matrix}s_{123}=12000&&s_{124}=6720&&s_{125}=8272&&s_{126}=-31584&&s_{134}=-37760,\\\
s_{135}=54784&&s_{136}=-35728&&s_{145}=-38080&&s_{146}=92208&&s_{156}=-30832&&\\\
s_{234}=-37920&&s_{235}=68288&&s_{236}=-108016&&s_{245}=-82896&&s_{246}=82416&&\\\
s_{256}=82720&&s_{345}=46592&&s_{346}=57664&&s_{356}=-19904&&s_{456}=-88944.\end{matrix}$

The scattering equations $(\ref{eq:scatter36})$ have $26$ solutions
$(x,y,z,w)\in\mathbb{C}^{4}$. Among these, we find:

$V$ | $W$ | $v(V\cap W^{\perp})$ | $v(V^{\perp}\cap W)$
---|---|---|---
$\bigl{(}\frac{8453}{5723},\frac{6083}{9263},\frac{3713}{1358},\frac{3713}{2198}\bigr{)}$ | $\bigl{(}\frac{6}{11},\frac{87}{172},-\frac{1}{4},-\frac{42}{43}\bigr{)}$ | $\bigl{(}\frac{5}{21},\frac{5}{36},\frac{19}{27},\frac{38}{69}\bigr{)}$ | $\bigl{(}\frac{6588}{14911},\frac{20988}{9139},-\frac{8601}{125060},\frac{17437}{138528}\bigr{)}$

Substituting these $(x,y,z,w)$ into (50), we obtain four $3\times 6$ matrices
$P$ with rational entries. These matrices represent the configurations
$V,\,W,\,v(V\cap W^{\perp}),\,v(V^{\perp}\cap W)$, where $V$ is the linear
span of the first three rows of $X$ and $W^{\perp}$ is the span of the last
three rows of $X$.

######  Conjecture 6.8.

The lifted scattering correspondence $\widetilde{\rm C}(3,n,1)$ decomposes
into five irreducible components, i.e. there is only one component whose map
onto $X(3,n)$ is not birational. Hence, the scattering correspondence ${\rm
C}(3,n,1)$ has three irreducible components. The four birational components
become two components modulo the involution in Remark 2.4.

The passage from the first sentence to the second sentence in Conjecture 6.8
mirrors the passage from Theorem 6.3 to Corollary 6.5. We verified our
conjecture in some small cases:

######  Proposition 6.9.

Conjecture 6.8 is true for $n=6,7,8$.

###### Proof.

The proof is a computation with HomotopyContinuation.jl [10]. Recall from [1]
that the degree of the map $\widetilde{\rm
C}(3,n,1)\rightarrow\mathrm{SH}(3,n,1)$ equals $26$, $1272$, $188112$ for
$n=6,7,8$. We ran an irreducible primary decomposition, based on monodromy
loops, on the defining equations of the lifted scattering correspondence. We
found five irreducible components over $X(3,n)$. For instance, for $n=7$, the
five components have degrees $1,1,1,1$ and $1268$. ∎

Acknowledgements: YEM is supported by Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) SFB-TRR 195 “Symbolic Tools in Mathematics and
their Application”. AP and BS are supported by the European Union (ERC,
UNIVERSE+, 101118787). We are grateful to Simon Telen for helping us with
numerical computations for Section 6.

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Authors’ addresses:

Yassine El Maazouz, RWTH Aachen and MPI MiS Leipzig yassine.el-
<EMAIL_ADDRESS>
Anaëlle Pfister, MPI MiS Leipzig<EMAIL_ADDRESS>
Bernd Sturmfels, MPI MiS Leipzig<EMAIL_ADDRESS>