# Equations of tensor eigenschemes

Valentina Beorchia, Francesco Galuppi and Lorenzo Venturello  Dipartimento di
Matematica e Geoscienze, Università di Trieste
via Valerio 12/1, 34126 Trieste, Italy<EMAIL_ADDRESS>Institute of
Mathematics of the Polish Academy of Sciences
Śniadeckich 8, 00-656 Warsaw, Poland<EMAIL_ADDRESS>Royal
Institute of Technology
Lindstedtsvg̈en 25, SE-100 44 Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract.

We study schemes of tensor eigenvectors from an algebraic and geometric
viewpoint. We characterize determinantal defining equations of such
eigenschemes via linear equations in their coefficients, both in the general
and in the symmetric case. We give a geometric necessary condition for a
0-dimensional scheme to be an eigenscheme.

###### 2020 Mathematics Subject Classification:

Primary 15A18, 14M12; Secondary 13P25, 15A69

##  1\. Introduction

Tensor spectral theory is an active area of research. Its applications range
from hypergraphs theory to quantum physics, from dynamical systems to
polynomial optimization, from signal processing to medical imaging. A
reference for these and many more applications is [13]. In this article we
study _eigenvectors of tensors_. There are several notions of tensor
eigenvectors as introduced independently in [12] and [15], all extending the
case of matrices. In the present paper we shall focus on the definition [16,
Section 2.2.1]. In this context it is not restrictive to consider partially
symmetric tensors of order $d$, that is on elements of $\mathop{\rm
Sym}\nolimits^{d-1}\mathbb{C}^{n+1}\otimes\mathbb{C}^{n+1}$.

If we choose a basis for $\mathbb{C}^{n+1}$, then we can identify a partially
symmetric tensor $T$ with a tuple $(g_{0},\dots,g_{n})$ of homogeneous
polynomials of degree $d-1$. With such a choice, an _eigenvector_ of $T$ is a
nonzero vector $v\in\mathbb{C}^{n+1}$ such that

(1) $(g_{0}(v),\ldots,g_{n}(v))=\lambda v$

for some constant $\lambda$. We observe that such vectors are called _E-
eigenvectors_ in [16, Section 2.2.1]. Since property (1) is preserved under
scalar multiplication, it is natural to regard eigenvectors as points in
$\mathbb{P}^{n}$; hence the name _eigenpoints_ instead of eigenvectors.

Motivated by [14, Question 16], we are interested in the geometry of
eigenpoints of tensors in the framework of complex projective geometry. One of
the reasons is that being an eigenpoint of a tensor is an algebraic condition,
described by the vanishing of minors of a suitable matrix - see Definition
2.1. Hence, eigenpoints of a tensor form a scheme.

There are several challenging questions about eigenschemes, their dimension,
their degree, their configurations and their defining equations. In this paper
we extend some of the results obtained in [3] in the case $n=2$ to arbitrary
dimension. For the convenience of the reader, we summarize here our main
contributions:

  * •

In 2.6 we explicitly compute the minimal graded free resolution of the
defining ideal of a general eigenscheme.

  * •

In 3.1 and 3.4 we characterize determinantal equations of eigenschemes of
tensors and symmetric tensors, thereby giving a positive solution to [3,
Conjecture 3.13]. These theorems allow us to design an algorithm which checks
if a tuple of homogeneous polynomials is the tuple of determinantal equations
of an eigenscheme. The second algorithm checks whether, given a set of points
$Z\subseteq\mathbb{P}^{n}$, there exists a tensor $T$, possibly symmetric,
whose eigenscheme contains $Z$. If the answer is positive, it explicitly
reconstructs $T$.

  * •

We prove that the variety parametrizing eigenschemes of symmetric tensors is
rational and we give a birational parametrization. This is the content of 4.3.

  * •

Finally, in 5.4 we give a geometric necessary condition for a $0$-dimensional
scheme to be the eigenscheme of a tensor. This gives a partial answer to [14,
Question 16].

######  Acknowledgment.

Beorchia (ORCID 0000-0003-3681-9045) is a member of GNSAGA of INdAM and is
supported by the fund Università degli Studi di Trieste - FRA 2022, by the
MIUR Excellence Department Project awarded to the DMG of Trieste, 2018-2023,
and by the Italian MIUR funds, PRIN project Moduli Theory and Birational
Classification (2017), P.i. U. Bruzzo.

Galuppi (ORCID 0000-0001-5630-5389) is supported by the National Science
Center, Poland, project “Complex contact manifolds and geometry of secants”,
2017/26/E/ST1/00231.

Venturello (ORCID 0000-0002-6797-5270) is supported by the Göran Gustafsson
foundation.

##  2\. Definitions and first results

In this section we define eigenschemes and discuss their algebraic invariants.

######  Definition 2.1.

Let $T=(g_{0},\ldots,g_{n})$ be a partially symmetric tensor. The
_eigenscheme_ of $T$ is the closed subscheme $E(T)\subseteq\mathbb{P}^{n}$
defined by the $2\times 2$ minors of

$\left(\begin{matrix}x_{0}&x_{1}&\dots&x_{n}\\\
g_{0}&g_{1}&\dots&g_{n}\end{matrix}\right).$

These minors are called _determinantal generators_ of the ideal $I_{E(T)}$.
When $T$ is symmetric, that is $T$ is identified with a homogeneous polynomial
$f$, we have $g_{i}=\partial_{i}f$ and we denote its eigenscheme by $E(f)$.

######  Remark 2.2.

The property of being an eigenvector is not invariant under linear
transformations. For this reason, when we speak of “the eigenpoints of a
tensor $T$” we are abusing terminology. In order to avoid ambiguities, we
choose once and for all a basis consisting of real vectors that are
orthonormal with respect to the Euclidean scalar product in
$\mathbb{R}^{n+1}$. With this choice we identify the space $\mathop{\rm
Sym}\nolimits^{d}\mathbb{C}^{n+1}$ of symmetric tensors with the space
$\mathbb{C}[x_{0},\dots,x_{n}]_{d}$ of degree $d$ homogeneous polynomials. In
the same way we identify $\mathop{\rm
Sym}\nolimits^{d-1}\mathbb{C}^{n+1}\otimes\mathbb{C}^{n+1}$ with $(\mathop{\rm
Sym}\nolimits^{d-1}\mathbb{C}^{n+1})^{\oplus(n+1)}$. In order to remember this
important choice, we always think of partially symmetric tensors as tuples of
polynomials. Once the basis is chosen, we can safely compute eigenvectors as
in (1).

It is possible to define the eigenscheme of a tensor which is not partially
symmetric, as in [4, Definition 1.1] or in [2, Section 1]. However, with the
choice of a basis described in Remark 2.2, it is apparent from the definition
that for every tensor $T$ there exists a partially symmetric tensor
$T^{\prime}$ such that $E(T^{\prime})=E(T)$. For this reason, it is not
restrictive to consider partially symmetric tensors.

######  Definition 2.3.

We define the integer $w(n,d)$ as

$w(n,d)=\frac{(d-1)^{n+1}-1}{d-2},$

for every $n\geq 1$ and $d\geq 3$. When $d=2$, we set $w(n,2)=n+1$.

The number $w(n,d)$ is the number of eigenpoints of a general tensor in
$(\mathop{\rm Sym}\nolimits^{d-1}\mathbb{C}^{n+1})^{\oplus(n+1)}$, as proven
in [4, Theorem 1.2].

######  Theorem 2.4.

Let $n\geq 1$, $d\geq 2$ and let $T\in(\mathop{\rm
Sym}\nolimits^{d-1}\mathbb{C}^{n+1})^{\oplus(n+1)}$ be a general tensor. Then
$E(T)\subseteq\mathbb{P}^{n}$ is a $0$-dimensional scheme of degree $w(n,d)$.

Since eigenschemes have an algebraic definition, it is natural to study
algebraic properties of their ideals. For instance, we are interested in their
minimal free resolution. In general this can be quite complicated, because
eigenschemes can have positive dimension and they can even have components of
different dimensions. However, the general tensor has a $0$-dimensional
eigenscheme, and the same is true for the general symmetric tensor - see [1,
Section 5.2]. For this reason we focus on the minimal graded free resolution
of the ideal of a $0$-dimensional eigenscheme. We accomplish that in Theorem
2.6.

Let $R=\mathbb{C}[x_{0},\dots,x_{n}]$, let $g_{0},\dots,g_{n}\in R_{d-1}$ and
consider the matrix

(2) $M=\begin{pmatrix}x_{0}&x_{1}&\dots&x_{n}\\\
g_{0}&g_{1}&\dots&g_{n}\end{pmatrix}.$

We regard $M$ as a map $R^{n+1}\to R(-1)\oplus R(-d+1)$ of free graded
$R$-modules. In order to study the ideal of minors of $M$, we recall the
definition of the Eagon-Northcott complex.

######  Definition 2.5.

Let $\alpha:F\to G$ a map of finitely generated $R$-modules, with

$f=\mathop{\rm rank}\nolimits(F)\mbox{ and }g=\mathop{\rm rank}\nolimits(G).$

The _Eagon-Northcott complex_ $EN(\alpha)_{\bullet}$ is the complex defined by

$\begin{cases}EN(\alpha)_{1}=\bigwedge^{g}G&\\\ EN(\alpha)_{i}=(\mathop{\rm
Sym}\nolimits_{i-1}(G))^{*}\otimes\bigwedge^{i+g-1}F&\mbox{ for every }2\leq
i\leq f-g+1,\end{cases}$

with differentials as given in [7, Section A2H].

In our setting $F=R^{n+1}$, $G=R(1)\oplus R(d-1)$ and $\alpha$ is the map
defined by left multiplication with $M$. The shifts are such that the map
$\alpha:F\to G$ with $(x_{0},\dots,x_{n})\mapsto M(x_{0},\dots,x_{n})^{T}$ is
homogeneous and they induce a shift in the modules $EN(\alpha)_{i}$.

######  Theorem 2.6.

Let $n$ and $d$ be positive integers such that $d\geq 2$ and let
$R=\mathbb{C}[x_{0},\dots,x_{n}]$. Let $I$ be the ideal generated by the
$2\times 2$ minors of a matrix $M$ as in (2) and let $V(I)$ be the zero locus
of $I$. If $\dim V(I)=0$, then $I$ is saturated and the minimal graded free
resolution

$F_{n}\to\ldots\to F_{1}\to I\to 0$

of $I$ as an $R$-module satisfies

$F_{i}\cong\bigoplus_{j=1}^{i}R(-j(d-2)-i-1)^{\binom{n+1}{i+1}}.$

###### Proof.

By hypothesis we have $\mathop{\rm codim}\nolimits V(I)=n=n+1-2+1$, so $V(I)$
is a determinantal subscheme of the expected codimension given by the
Porteous’ Formula - see [10, Corollary 11]. By the Hochster-Eagon Theorem -
see [9] or [6, Theorem 18.18] \- $R/I$ is a Cohen-Macaulay ring, so $I$ is
saturated. By [7, A2.10.C], the Eagon-Nothcott complex associated with
$\alpha$ is exact and it gives a minimal free resolution of $I$.

Specifically, in our setting we have

$\displaystyle(\mathop{\rm Sym}\nolimits_{i-1}(R$ $\displaystyle(1)\oplus
R(d-1)))^{*}=\bigoplus_{j=0}^{i-1}(\mathop{\rm
Sym}\nolimits_{i-1-j}(R(1)))^{*}\otimes(\mathop{\rm
Sym}\nolimits_{j}(R(d-1)))^{*}$
$\displaystyle=\bigoplus_{j=0}^{i-1}(\mathop{\rm
Sym}\nolimits_{i-1-j}(R)(i-1-j))^{*}(-1)\otimes(\mathop{\rm
Sym}\nolimits_{j}(R)(j(d-1)))^{*}$
$\displaystyle=\bigoplus_{j=0}^{i-1}(\mathop{\rm
Sym}\nolimits_{i-1-j}(R))^{*}(-i+1+j)\otimes(\mathop{\rm
Sym}\nolimits_{j}(R))^{*}(-j(d-1))$
$\displaystyle\cong\bigoplus_{j=0}^{i-1}R(-j(d-2)-i+1).$

The image of the map

$\bigwedge^{2}F\cong R^{\binom{n+1}{2}}\to\bigwedge^{2}G\cong R(d)$

is the ideal $I$ of $2\times 2$ minors of a matrix representing $\alpha$ as in
2.5. In our application, this is the defining ideal of the eigenscheme of a
partially symmetric tensor in $\mathop{\rm
Sym}\nolimits^{d-1}\mathbb{C}^{n+1}\otimes\mathbb{C}^{n+1}$. As we consider
this ideal in a standard graded polynomial ring $R$, we shift all the modules
in $EN(\alpha)$ by a factor of $-d$, i.e., we consider the complex
$EN(\alpha)_{\bullet}\otimes R(-d)$. Its $i$-th piece is equal to

$\displaystyle EN(\alpha)_{i}\otimes R(-d)$
$\displaystyle\cong\bigoplus_{j=0}^{i-1}R(-j(d-2)-i+1-d)^{\binom{n+1}{i+1}}$
$\displaystyle=\bigoplus_{j=0}^{i-1}R(-(j+1)(d-2)-i-1)^{\binom{n+1}{i+1}}.\qed$

######  Remark 2.7.

Notice that the defining ideal of an eigenscheme of a tensor, even of a
symmetric tensor, may fail to be saturated, as shown in [3, Remark 3.2]. In
this case the eigenscheme has dimension at least 1.

######  Remark 2.8.

The statement of Theorem 2.6 holds for any $0$-dimensional determinantal
scheme given by the $2\times 2$ minors of a $2\times(n+1)$ matrix whose first
row consists of linear entries and the second row of degree $d-1$ homogeneous
forms.

As a direct consequence we can see in the following Corollary that any matrix
whose maximal minors define a $0$-dimensional eigenscheme involves $n+1$
linearly independent linear entries in the first row. Indeed, by [11, Lemma
2.9], any such a matrix is of size $2\times(n+1)$, with homogeneous rows of
degrees $1$ and $d-1$. The next result is a generalization of the analogous
result for $n=2$ given in [2, Proposition 5.2].

######  Corollary 2.9.

Let $d\geq 2$. Let $l_{0},\dots,l_{n}\in\mathbb{C}[x_{0},\dots,x_{n}]_{1}$ and
$h_{0},\dots,h_{n}\in\mathbb{C}[x_{0},\dots,x_{n}]_{d-1}$. Let

$N=\begin{pmatrix}l_{0}&l_{1}&\dots&l_{n}\\\
h_{0}&h_{1}&\dots&h_{n}\end{pmatrix}$

and let $W\subseteq\mathbb{P}^{n}$ be the scheme defined by the $2\times 2$
minors of $N$. If $\dim W=0$, then $W$ is the eigenscheme of a tensor
$T\in(\mathbb{C}^{n+1})^{\otimes d}$ if and only if the linear forms
$l_{0},l_{1},\dots,l_{n}$ are linearly independent.

###### Proof.

If $l_{0},l_{1},\dots,l_{n}$ are linearly independent, then there exists
$B\in\mathop{\rm GL}\nolimits(n+1,\mathbb{C})$ such that

$N^{\prime}=N\cdot B=\begin{pmatrix}x_{0}&x_{1}&\dots&x_{n}\\\
b_{0}&b_{1}&\dots&b_{n}\end{pmatrix},$

where each $b_{j}\in\mathbb{C}[x_{0},\dots,x_{n}]_{d-1}$ is a linear
combination of the polynomials $h_{k}$. The zero set of the $2\times 2$ minors
of $N^{\prime}$ is $W$, and by definition it is the eigenscheme of the
partially symmetric tensor $(b_{0},b_{1},\dots,b_{n})$. Conversely, assume
that there exists a partially symmetric tensor $T=(g_{0},g_{1},\dots,g_{n})$
such that $E(T)=W$. As the $2\times 2$ minors of $N$ define a $0$-dimensional
eigenscheme, the Eagon-Northcott complex associated with $N$ is exact and. By
Remark 2.8, its first terms are given by

$R(-d-1)^{\binom{n+1}{3}}\oplus
R(-2d+1)^{\binom{n+1}{3}}\xrightarrow{d_{2}^{N}}R(-d)^{\binom{n+1}{2}}\to R$

By the definition of the first syzygy map $d_{2}^{N}:F_{2}^{N}\to F_{1}^{N}$,
it is possible to see that $d_{2}^{N}$ is defined by a matrix which is linear
in the entries of $N$. More precisely, by setting

$m_{ij}=l_{i}h_{j}-l_{j}h_{i},$

the generating syzygies are given by

$l_{i}m_{jk}-l_{j}m_{ik}+l_{k}m_{ij}=0\mbox{ and
}h_{i}m_{jk}-h_{j}m_{ik}+h_{k}m_{ij}=0$

for every $0\leq i<j<k\leq n$. In particular, the first $\binom{n+1}{3}$
columns of $d_{2}^{N}$ contain only the forms $l_{i}$ and $0$. Since $W$ is an
eigenscheme by assumption, its ideal is generated also by the $2\times 2$
minors of a matrix $M$ as in (2), so by Theorem 2.6 the Eagon-Northcott
complex associated with $M$ gives also a minimal free resolution of $I_{W}$.
By [6, Theorem 20.2], the minimal free resolution of a projective module is
unique up to isomorphism of complexes. This means that there is a matrix $H\in
M(\binom{n+1}{3},R_{d-2})$ with degree $d-2$ forms as entries, and there are
three invertible matrices $A\in\mathop{\rm
GL}\nolimits(\binom{n+1}{2},\mathbb{C})$ and $B,C\in\mathop{\rm
GL}\nolimits(\binom{n+1}{3},\mathbb{C})$ such that if we call

$G=\begin{pmatrix}B&\vline&H\\\ \hline\cr 0&\vline&C\end{pmatrix},$

then we have $d_{2}^{M}=A\cdot d_{2}^{N}\cdot G$. In particular, the linear
entries $x_{0},\dots,x_{n}$ of $d_{2}^{M}$ are linear combinations of
$l_{0},\dots,l_{n}$. So these forms span the whole of
$\mathbb{C}[x_{0},\dots,x_{n}]_{1}$, thus they are linearly independent. ∎

##  3\. Characterization of determinantal equations

In this section we give a characterization of tuples of homogeneous
polynomials which are determinantal generators of the homogeneous ideal of an
eigenscheme, as introduced in Definition 2.1.

When $n=2$, such a characterization has been found in [3, Section 3] for both
general and symmetric tensors. The main ingredients in our proof are two
prominent chain complexes in commutative algebra, namely the Koszul complex
and the de Rham complex. We begin with the general case.

######  Theorem 3.1.

Let $R=\mathbb{C}[x_{0},\dots,x_{n}]$. A tuple $(f_{ij}~{}:~{}0\leq i<j\leq
n)\in R_{d}^{\binom{n+1}{2}}$ of homogeneous polynomials is the tuple of
determinantal equations of the eigenscheme of a tensor if and only if

(3) $x_{i}f_{jk}-x_{j}f_{ik}+x_{k}f_{ij}=0$

for every $0\leq i<j<k\leq n$.

###### Proof.

By [6, Section 17.2], the Koszul complex of $x_{0},\dots,x_{n}$ starts with

$0\to R\xrightarrow{\alpha}R^{\oplus
n+1}\xrightarrow{\beta}\bigwedge^{2}R^{\oplus
n+1}\xrightarrow{\gamma}\bigwedge^{3}R^{\oplus n+1},$

where

$\displaystyle\alpha$ $\displaystyle(h)=x_{0}h+\ldots+x_{n}h,$
$\displaystyle\beta$
$\displaystyle(g_{0},\ldots,g_{n})=(x_{i}g_{j}-x_{j}g_{i}~{}:~{}0\leq i<j\leq
n)$ $\displaystyle\gamma$
$\displaystyle((f_{ij})_{i<j})=(x_{i}f_{jk}-x_{j}f_{ik}+x_{k}f_{ij}~{}:~{}0\leq
i<j<k\leq n).$

The tuple $(f_{ij}~{}:~{}0\leq i<j\leq n)$ is the tuple of determinantal
equations of the eigenscheme of a tensor if and only if
$(f_{ij})_{i<j}\in\mathop{\rm Im}\nolimits(\beta)$. Since $x_{0},\dots,x_{n}$
is regular sequence, the sequence is exact. In particular, $\mathop{\rm
Im}\nolimits(\beta)=\ker(\gamma)$. ∎

######  Remark 3.2.

We observe that exactness of the Koszul complex in the second term implies
that two partially symmetric tensors $T=(g_{0},\dots,g_{n})$ and
$T^{\prime}=(g_{0}^{\prime},\dots,g_{n}^{\prime})$ determine the same
determinantal equations for their eigenschemes if and only if there exist
$c\in\mathbb{C}^{*}$ and $h\in R_{d-2}$ such that

$g^{\prime}_{k}=cg_{k}+x_{k}h$

for every $k\in\\{0,\dots,n\\}$. Indeed, since $\mathop{\rm
Im}\nolimits(\alpha)=\ker(\beta)$, we have
$T^{\prime}-T=(x_{0}h,\dots,x_{n}h)$, for some $h\in R_{d-2}$.

As a consequence of the previous remark, we observe that the eigenschemes are
an example of general determinantal schemes, which admit a positive
dimensional family of defining matrices and they are not obtained by left and
right multiplication by a scalar invertible matrix. This is not the case for
general determinantal schemes of dimension $\geq 1$, which are defined by the
maximal minors of a matrix, as shown in [8, Main Theorem].

Now we are interested in a more refined version of Proposition 3.1. Namely, we
would like a similar characterization for determinantal equations of symmetric
tensors. In particular, we will prove [3, Conjecture 3.13]. Our argument
relies on the exactness of the affine de Rham complex, which we briefly
recall.

######  Definition 3.3.

Let $R=\mathbb{C}[x_{0},\dots,x_{n}]$. The _de Rham complex_ of
$\mathbb{A}^{n+1}$ is the complex of $R$-modules defined as

$0\to\Omega^{0}_{\mathbb{A}^{n+1}}\to\Omega^{1}_{\mathbb{A}^{n+1}}\to\cdots\to\Omega^{n+1}_{\mathbb{A}^{n+1}}\to
0,$

where $\Omega^{k}_{\mathbb{A}^{n+1}}$ is the module of algebraic differential
$k$-forms over $\mathbb{A}^{n+1}$. By letting $dx_{0},\dots,dx_{n}$ be an
$R$-basis for $\Omega^{1}_{\mathbb{A}^{n+1}}$ we can write a $k$-form
$\omega=\sum_{I\in\binom{[n+1]}{k}}f_{I}dx_{I}\in\Omega^{k}_{\mathbb{A}^{n+1}}$,
with $f_{I}\in R$ and with $dx_{I}=dx_{i_{1}}\wedge\cdots\wedge dx_{i_{k}}$
for $I=\\{i_{1},\dots,i_{k}\\}$. The differentials of the de Rham complex are
given by exterior derivatives, i.e.

$d\omega=\sum_{I\in\binom{[n+1]}{k}}\sum_{j=0}^{n}\partial_{j}f_{I}dx_{j}\wedge
dx_{I}.$

We are interested in the differential

$\displaystyle d:\Omega^{1}_{\mathbb{A}^{n+1}}$
$\displaystyle\to\Omega^{2}_{\mathbb{A}^{n+1}}$
$\displaystyle\sum_{i=0}^{n}f_{i}dx_{i}$
$\displaystyle\mapsto\sum_{i=0}^{n}\sum_{j=0}^{n}\partial_{j}f_{i}dx_{j}\wedge
dx_{i}=\sum_{0\leq i<j\leq n}(\partial_{i}f_{j}-\partial_{j}f_{i})dx_{i}\wedge
dx_{j}.$

As the de Rham complex of $\mathbb{A}^{n+1}$ is exact everywhere but in degree
0 (this follows, from instance, from de Rham’s theorem relating the cohomology
of de Rham complex with the singular cohomology of $\mathbb{A}^{n+1}$), we
conclude that whenever a tuple of $n+1$ polynomials $(f_{0},\dots,f_{n})$
satisfy the relations $\partial_{i}f_{j}-\partial_{j}f_{i}$ for every $0\leq
i<j\leq n$, then there exists an element of
$\Omega^{0}_{\mathbb{A}^{n+1}}\cong R$ such that $\nabla
g=(f_{0},\dots,f_{n})$. If moreover the polynomials $f_{0},\dots,f_{n}$ are
homogeneous of degree $d-1$, then by Euler formula $g$ is homogeneous of
degree $d$.

We are now in position to characterize determinantal generators of symmetric
tensors. The next result proves [3, Conjecture 3.13].

######  Theorem 3.4.

Let $d\geq 2$, $n\geq 2$ and let $R=\mathbb{C}[x_{0},\dots,x_{n}]$. A tuple
$(f_{ij}~{}:~{}0\leq i<j\leq n)\in R_{d}^{\binom{n+1}{2}}$ is the tuple of
determinantal equations of the eigenscheme of a polynomial in $R_{d}$ if and
only if

(4) $x_{i}f_{jk}-x_{j}f_{ik}+x_{k}f_{ij}=0$

and

(5) $\partial_{i}f_{jk}-\partial_{j}f_{ik}+\partial_{k}f_{ij}=0$

for every $0\leq i<j<k\leq n$.

Note that the equations labelled with (4) are precisely the equations in (3).

###### Proof.

It is immediate to check that the determinantal equations of the autoscheme of
a polynomial satisfy (4) and (5). Now we assume that the two equations hold
and we have to prove the existence of a polynomial $f\in R_{d}$ such that
$x_{i}\partial_{j}f-x_{j}\partial_{i}f=f_{ij}$. Assume that (4) holds for
every $(i,j,k)$. By 3.1 there exist polynomials $g_{0},\dots,g_{n}\in R_{d-1}$
such that $f_{ij}=x_{i}g_{j}-x_{j}g_{i}$ for every $0\leq i<j\leq n$. If (5)
holds for a fixed triple $(i,j,k)$, then

$\displaystyle 0$
$\displaystyle=\partial_{i}f_{jk}-\partial_{j}f_{ik}+\partial_{k}f_{ij}$
$\displaystyle=\partial_{i}(x_{j}g_{k}-x_{k}g_{j})-\partial_{j}(x_{i}g_{k}-x_{k}g_{i})+\partial_{k}(x_{i}g_{j}-x_{j}g_{i})$
(6)
$\displaystyle=x_{i}(-\partial_{j}g_{k}+\partial_{k}g_{j})-x_{j}(-\partial_{i}g_{k}+\partial_{k}g_{i})+x_{k}(-\partial_{i}g_{j}+\partial_{j}g_{i}).$

If $d=2$, then
$\deg(-\partial_{j}g_{k}+\partial_{k}g_{j})=\deg(-\partial_{i}g_{k}+\partial_{k}g_{i})=\deg(-\partial_{i}g_{j}+\partial_{j}g_{i})=0$.
The last equation above implies then that
$-\partial_{j}g_{k}+\partial_{k}g_{j}=-\partial_{i}g_{k}+\partial_{k}g_{i}=-\partial_{i}g_{j}+\partial_{j}g_{i}=0$.
By repeating the same step with other triples $(i,j,k)$ we obtain that
$\partial_{j}g_{i}-\partial_{i}g_{j}=0$ for every $0\leq i<j\leq n$. By what
we discussed above, this implies that there exists $f\in R_{d}$, such that
$\nabla f=(g_{0},\dots,g_{n})$. Substituting the identities
$\partial_{i}f=g_{i}$ in $f_{ij}=x_{i}g_{j}-x_{j}g_{i}$ we obtain that
$f_{ij}=x_{i}\partial_{j}f-x_{j}\partial_{i}f$ which proves the claim.
Assume the claim holds for every degree smaller than $d$. Then (6) shows that
the tuple $(\partial_{i}g_{j}-\partial_{j}g_{i}:0\leq i<j\leq n)\in R_{d-2}$
satisfy (4). By Schwartz theorem, it also satisfies (5). By induction, there
exists a polynomial $h\in R_{d-2}$ such that

$(\partial_{i}g_{j}-\partial_{j}g_{i})-(\partial_{i}x_{j}h-\partial_{j}x_{i}h)=\partial_{i}(g_{j}-x_{j}h)-\partial_{j}(g_{i}-x_{i}h)=0\\\
$

for every $0\leq i<j\leq n$. This implies the existence on a polynomial $f\in
R_{d}$ such that $g_{i}-x_{i}h=\partial_{i}f$. Since
$\partial_{i}x_{j}f-\partial_{j}x_{i}f=x_{j}(g_{i}-x_{i}h)-x_{i}(g_{j}-x_{j}h)=x_{j}g_{i}-x_{i}g_{j}=f_{ij}$
we conclude. ∎

3.1 and 3.4 give an effective algorithm to test whether $\binom{n+1}{2}$
homogeneous forms $h_{1},\dots,h_{\binom{n+1}{2}}$ of degree $d$ are the
defining equations of an eigenscheme (see Algorithm 1). Note that if
conditions (4) or (5) are not satisfied by the triples of polynomials $h_{i}$
in any order, we cannot conclude that these polynomials do not define the
eigenscheme of a tensor. Indeed, there might be a change of basis in the
linear system of the polynomials $h_{i}$ for which the new basis satisfies
conditions (4) or (5).

Input: $h_{1},\dots,h_{\binom{n+1}{2}}\in\mathbb{C}[x_{0},\dots,x_{n}]_{d}$.

Step 1: Set $M=(m_{ij})$ a $\binom{n+1}{2}\times\binom{n+1}{2}$ matrix of
indeterminates.

Step 2: Let $(f_{ij}~{}:~{}1\leq i<j\leq
n+1)=M\cdot(h_{1}\c{d}otsh_{\binom{n+1}{2}})^{\top}$. The coefficients of the
polynomials $f_{ij}$ are linear combinations of the $m_{ij}$.

Step 3: Consider the system of $\binom{n+1}{3}$ linear equations in the
$m_{ij}$ given by (3).

If the system has a nontrivial solutions

return False: $h_{1},\dots,h_{\binom{n+1}{2}}$ do not generate the eigenscheme
of a partially symmetric tensor.

Else
return True: $h_{1},\dots,h_{\binom{n+1}{2}}$ generate the eigenscheme of a
partially symmetric tensor.

Step 4: If Step 3 returned True, consider the system of $\binom{n+1}{3}$
linear equations in the $m_{ij}$ given by (5).

If the system has no nontrivial solution

return False: $h_{1},\dots,h_{\binom{n+1}{2}}$ do not generate the eigenscheme
of a symmetric tensor.

Else
return True: $h_{1},\dots,h_{\binom{n+1}{2}}$ generate the eigenscheme of a
symmetric tensor.

Algorithm 1 Test if homogeneous forms are defining equations of an eigenscheme

Another computational problem one is naturally concerned with is the
following: given $m$ distinct points in $\mathbb{P}^{n}$, is there an
eigenscheme containing them? In Algorithm 2 we propose a straightforward
procedure to test this, by simply considering a generic partially symmetric or
symmetric tensor and prescribing the vanishing of the minors of the associated
matrix (2) on the points. This yields a system of
$m\binom{n+1}{2}\binom{n+d}{d}$ linear equations in $(n+1)\binom{n+d-1}{d-1}$
(in the partially symmetric case) or $\binom{n+d}{d}$ (in the symmetric case)
variables. From a solution of this system, a witness tensor can be easily
reconstructed.

Input: $p_{1},\dots,p_{m}\in\mathbb{P}^{n}$, $d\geq 2$.

Step 1a: For $i\in\\{0,\dots,n\\}$, let

$g_{i}=\sum_{\alpha}\lambda_{i,\alpha}x^{\alpha},$

where $x^{\alpha}$ ranges among all degree $d-1$ monomials in the variables
$x_{0},\ldots,x_{n}$. In words, each $g_{i}$ is a polynomial of degree $d-1$
with distinct unknowns $\lambda_{i,\alpha}$ as coefficients.

Step 2a: Consider the system of $m\binom{n+1}{2}\binom{n+d}{d}$ linear
equations in the variables $\lambda_{i,\alpha}$ given by
$x_{i}g_{j}(p_{k})-x_{j}g_{i}(p_{k})=0$ for every $0\leq i<j\leq n$ and for
every $k\in\\{1,\dots,m\\}$.

If the system has a non trivial solution

return  True: $p_{1},\dots,p_{m}$ are contained in the eigenscheme of a
partially symmetric tensor in
$\text{Sym}^{d-1}(\mathbb{C}^{n+1})\otimes\mathbb{C}^{n+1}$. A witness tensor
is given by $(g_{0},\dots,g_{n})$, after evaluating the coefficients of each
$g_{i}$ in a solution of the system.

Step 1b: Let

$f=\sum_{\alpha}\lambda_{i,\alpha}x^{\alpha},$

where $x^{\alpha}$ ranges among all degree $d$ monomials in the variables
$x_{0},\ldots,x_{n}$.

Step 2b: Consider the system of $m\binom{n+1}{2}\binom{n+d}{d}$ linear
equations in the variables $\lambda_{i,\alpha}$ given by
$x_{i}\partial_{j}f(p_{k})-x_{j}\partial_{i}f(p_{k})=0$ for every $0\leq
i<j\leq n$ and for every $k\in\\{1,\dots,m\\}$.

If the system has a non trivial solution

return  True: $p_{1},\dots,p_{m}$ are contained in the eigenscheme of a
symmetric tensor in $\text{Sym}^{d}(\mathbb{C}^{n+1})$. A witness tensor is
given by $f$, after evaluating the coefficients in a solution of the system.

Algorithm 2 Test if points are contained in an eigenscheme of a tensor of
fixed degree

##  4\. The variety of eigenpoints

Now we take a more geometric approach and we turn our attention to the variety
parametrizing configurations of eigenpoints of tensors. Let
$(\mathbb{P}^{n})^{w(n,d)}$ be the symmetric power of $\mathbb{P}^{n}$ with
itself $w(n,d)$ times, where $w(n,d)$ is the integer in Definition 2.3. In
other words, $(\mathbb{P}^{n})^{w(n,d)}$ parametrizes sets of $w(n,d)$ points
in $\mathbb{P}^{n}$. Since $w(n,d)$ is precisely the degree of the eigenscheme
of a generic tensor, following [2, Section 5] we define the variety of
eigenconfigurations $\mathop{\rm
Eig}\nolimits_{n,d}\subseteq(\mathbb{P}^{n})^{w(n,d)}$ to be the Zariski
closure of the set of all eigenconfigurations of tensors in
$(\mathbb{C}^{n+1})^{\otimes d}$.

The dimension of $\mathop{\rm Eig}\nolimits_{2,d}$ has been determined in [2,
Theorem 5.5]. Thanks to Remark 3.2, we can give an upper bound for such a
dimension in general.

######  Proposition 4.1.

For any $n\geq 2$ and $d\geq 3$ we have

$\dim\mathop{\rm
Eig}\nolimits_{n,d}\leq\dfrac{n(n+d)}{d-1}\binom{n+d-2}{n}-1.$

###### Proof.

Consider the dominant rational map

$\eta_{n,d}:\mathbb{P}(\mathbb{C}[x_{0},\dots,x_{n}]_{d-1})\times\dots\times\mathbb{P}(\mathbb{C}[x_{0},\dots,x_{n}]_{d-1})\dasharrow\mathop{\rm
Eig}\nolimits_{n,d},$

with $\eta_{n,d}([g_{0}],\dots,[g_{n}])=E(T)$ and $T=(g_{0},\dots,g_{n})$. By
Remark 3.2, the general fiber of $\eta_{n,d}$ contains a copy of
$\mathbb{P}(\mathbb{C}[x_{0},\dots,x_{n}]_{d-2})$. Since the (affine)
dimension of $\mathop{\rm Eig}\nolimits_{n,d}$ is equal to the dimension of
the source of the map $\eta_{n,d}$ minus that of the generic fiber, we have
projectively that

$\dim\mathop{\rm
Eig}\nolimits_{n,d}\leq(n+1)\binom{n+d-1}{n}-\binom{n+d-2}{n}-1,$

and the claim follows. ∎

By [2, Theorem 5.5] we have $\dim\mathop{\rm Eig}\nolimits_{2,d}=d^{2}+2d-1$,
so the bound given by 4.1 is sharp for $n=2$. We expect equality to hold for
every $n\geq 2$, but at the moment we lack a characterization of all possible
defining matrices for a given $0$-dimensional eigenscheme.

Now we turn to the symmetric case. The dimension of the corresponding variety
has already been computed in [17, Theorem 1.1]. Here we prove that
$\mathop{\rm Eig}\nolimits_{n,d,\mathop{\rm Sym}\nolimits}$ is rational and
give a birational parametrization. Our argument is similar to the one
presented in [3, Remark 3.7].

######  Definition 4.2.

We set

$\begin{matrix}\phi_{n,d}:&\mathbb{P}(\mathbb{C}[x_{0},\ldots,x_{n}]_{d})&\dashrightarrow&(\mathbb{P}^{n})^{w(n,d)}\\\
&f&\mapsto&E(f),\end{matrix}$

and we define $\mathop{\rm Eig}\nolimits_{n,d,\mathop{\rm Sym}\nolimits}$ to
be the closure of the image of $\phi_{n,d}$.

Using [4, Theorem 1.2], we obtain that $\phi_{n,d}$ is defined on all forms
$f$ such that $E(f)$ is reduced and $0$-dimensional.

Let $\alpha$ be the projectivization of the linear map

$\begin{matrix}\mathbb{C}[x_{0},\ldots,x_{n}]_{d}&\to&(\mathbb{C}[x_{0},\ldots,x_{n}]_{d})^{\oplus\binom{n+1}{2}}\\\
f&\mapsto&(x_{i}\partial_{j}f-x_{j}\partial_{i}f)_{i<j},\end{matrix}$

and define

$T_{n,d}=\overline{\alpha(\mathbb{P}(\mathbb{C}[x_{0},\ldots,x_{n}]_{d}))}\subseteq\mathbb{P}((\mathbb{C}[x_{0},\ldots,x_{n}]_{d})^{\oplus\binom{n+1}{2}})$

the closure of the image of $\alpha$. Then $T_{n,d}$ is a linear subspace of
$\mathbb{P}((\mathbb{C}[x_{0},\ldots,x_{n}]_{d})^{\oplus\binom{n+1}{2}})$ and
there is a commutative diagram

$\textstyle{\mathbb{P}(\mathbb{C}[x_{0},\ldots,x_{n}]_{d})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{n,d}}$$\scriptstyle{\alpha}$$\textstyle{\mathop{\rm
Eig}\nolimits_{n,d,\mathop{\rm
Sym}\nolimits}}$$\textstyle{T_{n,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{n,d}}$

where $\psi_{n,d}(f_{ij}~{}:~{}0\leq i<j\leq n)$ is the set of points defined
by the ideal $(f_{ij}~{}:~{}i<j)$ in $\mathbb{P}^{n}$.

######  Proposition 4.3.

Let $n$ and $d$ be positive integers such that $d\geq 3$. The map
$\psi_{n,d}:T_{n,d}\dashrightarrow\mathop{\rm Eig}\nolimits_{n,d,\mathop{\rm
Sym}\nolimits}$ is birational. In particular, $\mathop{\rm
Eig}\nolimits_{n,d,\mathop{\rm Sym}\nolimits}$ is an irreducible, rational
variety.

###### Proof.

We distinguish two cases. If $d$ is odd, then $\alpha$ is an isomorphism by
[3, Lemma 2.8]. Since $\phi_{n,d}$ is birational by [17, Theorem 1.1],
$\psi_{n,d}$ is birational as well. Assume now that $d$ is even. The fibers of
$\alpha$ are lines by [3, Lemma 2.8] and the general fiber of $\phi_{n,d}$ is
a line by [17, Theorem 1.1]. This implies that $\psi_{n,d}$ is generically
injective. ∎

######  Remark 4.4.

Another description of the eigenvariety was found in [5, Section 5]. The
authors show that $\mathop{\rm Eig}\nolimits_{3,3,\mathop{\rm Sym}\nolimits}$
is birational to a linear section of the Grassmanian $\mathbb{G}(3,14)$, but
their argument can be easily adapted to different values of $n$ and $d$.

##  5\. Generalized Laguerre maps

The goal of this section is to give a necessary condition for a
$0$-dimensional scheme to be the eigenscheme of a tensor. We accomplish this
task in Proposition 5.4. Our argument relies on [3, Section 5] and consists in
studying the fibers of the Laguerre map associated with a partially symmetric
tensor.

######  Definition 5.1.

Let $n\geq 2$ and $d\geq 3$, and let $T=(g_{0},\dots,g_{n})\in\mathop{\rm
Sym}\nolimits^{d-1}\mathbb{C}^{n+1}\otimes\mathbb{C}^{n+1}$ be a partially
symmetric tensor. For every $0\leq i<j\leq n$, let
$f_{ij}=x_{i}g_{j}-x_{j}g_{i}$ be the $2\times 2$ minors of the matrix (2).
The Laguerre map associated with $T$ is the map

$\lambda_{T}:\mathbb{P}^{n}\dasharrow\mathbb{P}(\Lambda^{2}\mathbb{C}^{n+1})$

given by $\lambda_{T}(P)=(f_{ij}(P):0\leq i<j\leq n)$. It is defined on the
open subset $\mathbb{P}^{n}\setminus E(T)$.

######  Remark 5.2.

Let us take a closer look at the image of $\lambda_{T}$. It is easy to see
that
$\lambda_{T}(\mathbb{P}^{n})\subsetneq\mathbb{P}(\Lambda^{2}\mathbb{C}^{n+1})$.
Indeed, the coordinates of $\lambda_{T}(P)$ are the Plücker coordinates of the
line in $\mathbb{P}^{n}$ joining $P$ and $(g_{0}(P):\ldots:g_{n}(P))$.
Therefore $\lambda_{T}(\mathbb{P}^{n})\subseteq\mathbb{G}(1,n)$.

By definition, the ideal of $E(T)$ is generated by the polynomials $f_{ij}$,
hence the closure of the graph of $\lambda_{T}$ is a model of the blow-up
${\mathop{\rm
Bl}\nolimits}_{E(T)}\mathbb{P}^{n}\subseteq\mathbb{P}^{n}\times\mathbb{G}(1,n)$.
To determine its equations, observe that a point of $\mathbb{G}(1,n)$ with
Plücker coordinates corresponding to the $2\times 2$ minors of a rank 2 matrix

$\left(\begin{matrix}a_{0}&a_{1}&\dots&a_{n}\\\ b_{0}&b_{1}&\dots&b_{n}\\\
\end{matrix}\right)$

lies in the image of $\lambda_{T}$ if and only if there exists a point
$P=(P_{0}:P_{1}:\ldots:P_{n})\in\mathbb{P}^{n}$ such that

$\mathop{\rm rank}\nolimits\left(\begin{matrix}a_{0}&a_{1}&\dots&a_{n}\\\
b_{0}&b_{1}&\dots&b_{n}\\\ P_{0}&P_{1}&\dots&P_{n}\\\
g_{0}(P)&g_{1}(P)&\dots&g_{n}(P)\end{matrix}\right)=2.$

Therefore the equations of ${\mathop{\rm Bl}\nolimits}_{E(T)}\mathbb{P}^{n}$
in $\mathbb{P}^{n}\times\mathbb{G}(1,n)$ are given by the $3\times 3$ minors.
Since the first two rows are linearly independent, by Kronecker’s theorem on
the rank of a matrix it is enough to consider the $3\times 3$ minors obtained
by deleting the third row and the $3\times 3$ minors obtained deleting the
fourth row. In particular, there are $\binom{n+1}{3}$ equations of bidegree
$(1,1)$ and $\binom{n+1}{3}$ equations of bidegree $(d-1,1)$.

We now want to study the fibers of $\lambda_{T}$. Let

$p_{1}:{\mathop{\rm
Bl}\nolimits}_{E(T)}\mathbb{P}^{n}\to\mathbb{P}^{n}~{}\mbox{ and
}~{}p_{2}:{\mathop{\rm Bl}\nolimits}_{E(T)}\mathbb{P}^{n}\to\mathbb{G}(1,n)$

be the projections. We observe that
$\lambda_{T}^{-1}(\omega)=p_{1}(p_{2}^{-1}(\omega))$ for every
$\omega\in\mathbb{G}(1,n)$. Hence it suffices to look at fibers of $p_{2}$.

######  Proposition 5.3.

Let $T=(g_{0},\dots,g_{n})\in\mathop{\rm
Sym}\nolimits^{d-1}\mathbb{C}^{n+1}\otimes\mathbb{C}^{n+1}$ be a partially
symmetric tensor. Then every fiber of
$\lambda_{T}:\mathbb{P}^{n}\dasharrow\mathbb{G}(1,n)$ is contained in a line
in $\mathbb{P}^{n}$.

###### Proof.

Let $\omega\in\mathop{\rm Im}\nolimits(\lambda_{T})\subseteq\mathbb{G}(1,n)$.
Then there exist $Q_{0},\ldots,Q_{n}\in\mathbb{C}$ such that the homogeneous
coordinates of $\omega$ are the $2\times 2$ minors of the matrix (2) evaluated
at $Q=(Q_{0}:\ldots:Q_{n})$. In other words, the Plücker coordinates of
$\omega$ are $(\bar{p}_{ij}~{}:~{}0\leq i<j\leq n)$, where
$\bar{p}_{ij}=Q_{i}g_{j}(Q)-Q_{j}g_{i}(Q)\in\mathbb{C}$. The fiber
$p_{2}^{-1}(\omega)$ is contained in the subvariety of
$\mathbb{P}^{n}\times\mathbb{G}(1,n)$ whose equations in $P_{0},\dots,P_{n}$
are the $3\times 3$ minors of the matrix

$\left(\begin{matrix}Q_{0}&Q_{1}&\dots&Q_{n}\\\
g_{0}(Q)&g_{1}(Q)&\dots&g_{n}(Q)\\\ P_{0}&P_{1}&\dots&P_{n}\\\
\end{matrix}\right).$

For all $0\leq i<j<k\leq n$, we get

(7) $\bar{p}_{ij}P_{k}-\bar{p}_{ik}P_{j}+\bar{p}_{jk}P_{i}=0.$

Once we project on the first factor, we see that the fiber
$\lambda_{T}^{-1}(\omega)=p_{1}(p_{2}^{-1}(\omega))\subseteq\mathbb{P}^{n}$ is
contained in the linear subspace defined by the $\binom{n+1}{3}$ linear
equations (7) in the variables $P_{0},\dots,P_{n}$. It is not difficult to see
that the coefficient matrix $A_{\omega}$ of this linear system gives the
linear map $\mathbb{C}^{n+1}\to\bigwedge^{3}\mathbb{C}^{n+1}$ defined by

$A_{\omega}(v)=\omega\wedge v.$

For instance, for $n=3$ we get

$A_{\omega}=\left(\begin{matrix}\bar{p}_{12}&-\bar{p}_{02}&\bar{p}_{01}&0\\\
\bar{p}_{13}&-\bar{p}_{03}&0&\bar{p}_{01}\\\
\bar{p}_{23}&0&-\bar{p}_{03}&\bar{p}_{02}\\\
0&\bar{p}_{23}&-\bar{p}_{13}&\bar{p}_{12}\\\ \end{matrix}\right).$

By recalling that a 2-vector $\omega$ is decomposable if and only if $\dim\ker
A_{\omega}=2$, we have that $\mathop{\rm rank}\nolimits A_{\omega}=n-1$. It
follows that $\lambda_{T}^{-1}(\omega)$ is contained in an intersection of
$n-1$ hyperplanes in $\mathbb{P}^{n}$, which is a line. ∎

Equations (7) determine the projectivization of the tautological bundle of
$\mathbb{G}(1,n)$ inside $\mathbb{P}^{n}\times\mathbb{G}(1,n)$.

Proposition 5.3 allows us to prove the main result of this section, thereby
generalizing [3, Theorem 5.5].

######  Proposition 5.4.

Let $n\geq 2$, $d\geq 2$ and let $T\in(\mathbb{C}^{n+1})^{\otimes d}$ be a
tensor such that $\dim E(T)=0$. Then for every $k\in\\{2,\dots,d-1\\}$, no
degree $kd$ subscheme of $E(T)$ is contained in an irreducible curve of degree
$k$. Moreover, no degree $d+1$ subscheme of $E(T)$ is contained in a line.

###### Proof.

As usual, we can assume that $T\in\mathop{\rm
Sym}\nolimits^{d-1}\mathbb{C}^{n+1}\otimes\mathbb{C}^{n+1}$ is partially
symmetric. In [3, Proposition 2.6(1)] we can see how $E(T)$ cannot contain
$d+1$ collinear points, and the argument holds verbatim for a $0$-dimensional
scheme of degree $d+1$. Assume that there exists a degree $kd$ subscheme of
$E(T)$ contained in an irreducible curve $C\subseteq\mathbb{P}^{n}$ of degree
$k$. If we call $\mathcal{L}=\langle f_{ij}~{}:~{}0\leq i<j\leq n\rangle$ the
linear system on $\mathbb{P}^{n}$ associated with the Laguerre map
$\lambda_{T}$, then we can interpret
$\lambda_{T}:\mathbb{P}^{n}\dashrightarrow\mathbb{P}(\mathcal{L}^{*})$ as the
map assigning to a point $P\in\mathbb{P}^{n}\setminus E(T)$ the hyperplane

$\mathcal{L}_{P}=\\{D\in\mathcal{L}~{}:~{}P\in D\\}\subseteq\mathcal{L}.$

If $P\in C$, then every $D\in\mathcal{L}_{P}$ contains $E(T)\cup\\{P\\}$, so
it intersects $C$ with multiplicity at least $kd+1>\deg(C)\cdot\deg(D)$. Since
$C$ is irreducible, we deduce that $C\subseteq D$ by Bézout’s Theorem. Hence

$\lambda_{T}(P)=\mathcal{L}_{P}=\\{D\in\mathcal{L}~{}:~{}C\subseteq D\\}$

is constant for every $P\in C\setminus E(T)$. It follows that $C$ is
contracted by $\lambda_{T}$, and this implies that $k=1$ by Proposition 5.3. ∎

The bound on collinear points given by Proposition 5.4 is sharp. Indeed, it is
easy to find order $d$ tensors whose eigenscheme contains $d$ points on a
line, see for instance [5, Example 4.2].

## References

  * [1] Abo, H., On the discriminant locus of a rank $n-1$ vector bundle on $\mathbb{P}^{n-1}$, Portugaliae Mathematica, 77, (2020), 3-4, 299–343.
  * [2] H. Abo and A. Seigal and B. Sturmfels, Eigenconfigurations of Tensors, Algebraic and Geometric Methods in Discrete Mathematics, Contemporary Mathematics 685, AMS, Providence, RI, (2017), 1–25.
  * [3] Beorchia, V. and Galuppi, F. and Venturello, L., Eigenschemes of ternary tensors, SIAM Journal on Applied Algebra and Geometry, 5, (2021), 4, 620–650.
  * [4] Cartwright, D. and Sturmfels, B., The number of eigenvalues of a tensor, Linear Algebra and its Applications, 438, (2013), 2, 942–952.
  * [5] Çelik, T. Ö. and Galuppi, F. and Kulkarni, A. and Sorea, M.-Ş., On the eigenpoints of cubic surfaces, Matematiche (Catania), 75, (2020).
  * [6] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, (1995).
  * [7] Eisenbud, D., The geometry of syzygies, Graduate Texts in Mathematics, 229, Springer-Verlag, New York, (2005).
  * [8] Faenzi, D. and Fania, M. L., On the Hilbert scheme of varieties defined by maximal minors, Math. Res. Lett., 21, (2014), 2, 297–311.
  * [9] Hochster, M. and Eagon, John A., Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math., 93, (1971), 1020–1058.
  * [10] Kempf, G. and Laksov, D., The determinantal formula of Schubert calculus, Acta Math., 132, (1974), 153–162.
  * [11] Kreuzer, M. and Migliore, J. C. and Peterson, C. and Nagel, U., Determinantal schemes and Buchsbaum-Rim sheaves, J. Pure Appl. Algebra, 150, (2000), 2, 155–174.
  * [12] Lim, L.-H. , Singular values and eigenvalues of tensors: a variational approach, 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, (2005), 129–132.
  * [13] Qi, L. and Chen, H. and Chen, Y., Tensor eigenvalues and their applications, Advances in Mechanics and Mathematics, 39, Springer, Singapore, (2018).
  * [14] Ranestad, K. and Sturmfels, B., Twenty-seven questions about the cubic surface, Matematiche (Catania), 75, (2020), 2, 411–424.
  * [15] Qi, L., Eigenvalues of a real supersymmetric tensor, Journal of Symbolic Computation, 40, (2005), 6, 1302–1324.
  * [16] Qi, L. and Luo, Z., Tensor analysis: Spectral theory and special tensors, Philadelphia, PA: Society for Industrial and Applied Mathematics, (2017).
  * [17] Turatti, E., On tensors that are determined by their singular tuples, preprint arXiv 2104:03686, (2021).