#  Self-dual toric varieties

Apostolos Thoma, Marius Vladoiu  Apostolos Thoma, Department of Mathematics,
University of Ioannina, Ioannina 45110, Greece<EMAIL_ADDRESS>Marius Vladoiu,
Faculty of Mathematics and Computer Science, University of Bucharest, Str.
Academiei 14, Bucharest, RO-010014, Romania<EMAIL_ADDRESS>
###### Abstract.

We describe explicitly all multisets of weights whose defining projective
toric varieties are self-dual. In addition, we describe a remarkable and
unexpected combinatorial behaviour of the defining ideals of these varieties.
The toric ideal of a self-dual projective variety is weakly robust, that means
the Graver basis is the union of all minimal binomial generating sets. When,
in addition, the self-dual projective variety has a non-pyramidal
configuration, then the toric ideal is strongly robust, namely the Graver
basis is a minimal generating set, therefore there is only one minimal
binomial generating set which is also a reduced Gröbner basis with respect to
every monomial order and thus, equals the universal Gröbner basis.

###### Key words and phrases:

toric varieties, self dual, Gale transforms, Markov basis, Graver basis,
Gröbner basis

###### 1991 Mathematics Subject Classification:

14M25, 13P10, 52B35, 14N05

Corresponding author: Apostolos Thoma

##  1\. Introduction

Let $K$ be an algebraically closed field of characteristic zero and $A$ be an
$m\times n$ integer matrix such that $(1,1,\cdots,1,1)$ belongs in the row
span of the matrix $A$. For a vector ${\bf
u}\in\operatorname{Ker}_{{\mathbb{Z}}}(A)$, we denote by ${\bf u}^{+},{\bf
u}^{-}\in{\mathbb{N}}^{n}$, the unique vectors of disjoint support such that
${\bf u}={\bf u}^{+}-{\bf u}^{-}$. The support of ${\bf u}$ is the set
$\\{i|u_{i}\neq 0\\}$. The toric ideal of $A$ is the homogeneous ideal
$I_{A}\subset K[x_{1},\ldots,x_{n}]$ generated by the binomials
${x}^{{{\mathbb{u}}}^{+}}-{x}^{{{\mathbb{u}}}^{-}}$ where
${{\mathbb{u}}}\in\operatorname{Ker}_{{\mathbb{Z}}}(A),$ see [26, Chapter 4].
Let $T=(K^{*})^{m}$ be the algebraic torus of rank $m$ over $K$ and $\chi(T)$
be the lattice of characters of $T$; characters are of the form $\lambda({\bf
t})={\bf t}^{\bf b}=t_{1}^{b_{1}}t_{2}^{b_{2}}\cdots t_{m}^{b_{m}}$, where
${\bf b}\in{\mathbb{Z}}^{m}$, thus we can identify $\chi(T)$ with
${\mathbb{Z}}^{m}$. Then $T$ acts on $K^{n}$ by ${\bf
t}\cdot(x_{1},x_{2},\ldots,x_{n})=({\bf t}^{{\bf a}_{1}}x_{1},{\bf t}^{{\bf
a}_{2}}x_{2},\ldots,{\bf t}^{{\bf a}_{n}}x_{n})$, where the set of weights
${\bf a}_{1},{\bf a}_{2},\ldots,{\bf a}_{n}$ are the columns of the matrix
$A$. The projective toric variety
$X_{A}=V(I_{A})\subset{\mathbb{P}}_{K}^{n-1}$ is the Zariski closure of the
orbit of the point $(1,1,\ldots,1,1)$. By $X_{A}^{*}$ we denote the dual
variety of $X_{A}$, which is the closure in the dual projective space of the
hyperplanes intersecting the regular points of $X_{A}$ non transversally. The
variety $X_{A}$ is called self-dual if it is isomorphic to its dual
$X_{A}^{*}$ as embedded projective varieties. In [5], Bourel, Dickenstein and
Rittatore prove that the variety $X_{A}$ is a self-dual projective toric
variety if and only if $\dim(X_{A})=\dim(X_{A}^{*})$ and the smallest linear
subspaces containing $X_{A}$ and $X_{A}^{*}$ have the same dimension. The
variety $X_{A}$ is called defective if the codimension of its dual variety is
not one. Thus self-dual toric varieties which are not hypersurfaces are
defective. The classification of defective projective toric varieties is a
difficult problem with a lot of recent interest, see [13, 12, 16, 17, 6].

Also, in [5], Bourel et al. produce the full list of self-dual smooth
projective toric varieties and several examples of non-smooth self dual toric
vatieties. They also provide combinatorial characterizations of self-duality
for toric varieties. In this article, we combine the combinatorial
characterizations of [5] and the full bouquet theory, including the
generalized Lawrence matrices [24], which permit us to describe explicitly all
multisets of weights whose defining projective toric varieties are self-dual
and we prove their interesting algebro-combinatorial properties. Toric ideals
of self-dual projective toric varieties associated to non-pyramidal
configurations posess the same combinatorial properties [26, Proposition 7.1]
as the toric ideals of the second Lawrence liftings: they are strongly robust,
see [28]. A toric ideal is called robust if it is minimally generated by its
universal Gröbner basis, where the universal Gröbner basis is the union of all
reduced Gröbner bases, see [3]. Robust related properties of ideals are
studied in several recent articles; see [1, 2, 10, 11, 27] for robust ideals,
[3, 4] for robust toric ideals, [19, 21, 22, 24, 25, 26, 28] for strongly
robust toric ideals and [18, 29] for generalized robust toric ideals.

A strongly robust toric ideal is a toric ideal $I_{A}$ for which the Graver
basis $\operatorname{Gr}(I_{A})$ is a minimal system of generators, see [28].
Since $(1,1,\cdots,1,1)$ belongs to the row span of the matrix $A$, then
$\operatorname{Ker}_{{\mathbb{Z}}}(A)\cap{\mathbb{N}}^{n}=\\{{\mathbf{0}}\\}$,
which implies that any minimal binomial generating set is contained in the
Graver basis, see [9, Theorem 2.3]. Then, for strongly robust ideals, the
Graver basis is the unique minimal system of generators and, thus, any reduced
Gröbner basis as well as the universal Gröbner basis are identical with the
Graver basis, since all of them contain a minimal system of generators and
they are subsets of the Graver basis (see [26, Chapter 4]). We conclude that
for a strongly robust toric ideal $I_{A}$ the following sets are identical:
the set of indispensable elements, any minimal system of binomial generators,
any reduced Gröbner basis, the universal Gröbner basis and the Graver basis.
Therefore, strongly robust toric ideals are robust. The set of indispensable
elements is the intersection of all minimal binomial sets of generators, while
their union is the universal Markov basis, up to the identification of
opposite binomials. Weakly robust are the toric ideals for which the universal
Markov basis and the Graver basis of $I_{A}$ coincide. The ideals of self-dual
toric varieties, pyramidal or non-pyramidal, are weakly robust.

In Section 2, we present the basic theory of bouquets. In Section 3, we make
an introduction to the generalized Lawrence matrices and we use them to
describe explicitly all multisets of weights whose defining projective toric
varieties are self-dual. Finally, in Section 4, we present the necessary
notions from the different toric bases and we prove that the toric ideal of a
self-dual projective variety is weakly robust. When, in addition, the self-
dual projective variety is defined by a non-pyramidal configuration, then the
toric ideal is strongly robust.

##  2\. Preliminaries on bouquets

Let $\\{{\bf a}_{1},\dots,{\bf a}_{n}\\}\subset{\mathbb{Z}}^{m}$. Throughout
the article we will use the same symbol, $A$, for the set $\\{{\bf
a}_{1},\dots,{\bf a}_{n}\\}$ and for the integer matrix $[{\bf
a}_{1},\dots,{\bf a}_{n}]\in{\mathbb{Z}}^{m\times n}$ with columns ${\bf
a}_{1},\dots,{\bf a}_{n}$.

To any integer matrix $A\in{\mathbb{Z}}^{m\times n}$ one can associate an
oriented matroid structure consisting of a graph $G_{A}$ and another matrix
$A_{B}$, called the “bouquet graph” and the “bouquet matrix”, respectively.
The name bouquet was inspired by the article [23] and the theory of bouquets
was developed in a series of articles [24, 25, 21]. But a lot of the ideas
behind the bouquet structure were used extensively before the definition to
study the self-dual projective toric varieties in [5].

The bouquet graph $G_{A}$ of $I_{A}$ is the graph on the set of vertices
$\\{{\bf a}_{1},\dots,{\bf a}_{n}\\}$ and edges
$\\{{\mathbb{a}}_{i},{\mathbb{a}}_{j}\\}$ if there exists a vector
${\mathbb{v}}\in{\mathbb{Z}}^{m}$ such that the dot products
${\mathbb{v}}\cdot{\mathbb{a}}_{k}=0$ for all $k\neq i,j$ and
${\mathbb{v}}\cdot{\mathbb{a}}_{i}\neq 0$,
${\mathbb{v}}\cdot{\mathbb{a}}_{j}\neq 0$ or equivalently that there is a
covector with support $i,j$ in the vector matroid defined by the columns of
$A$, see [24, Definition 1.1]. The bouquets are the connected components of
the graph $G_{A}$.

The bouquet graph can also be defined in terms of the Gale transform of the
matrix $A$. To any matrix $A$ one can associate its Gale transform, which is
the $n\times(n-r)$ matrix whose columns span the lattice
$\operatorname{Ker}_{{\mathbb{Z}}}(A)$, where $r$ is the rank of $A$. We will
denote the set of ordered row vectors of the Gale transform of $A$ by
$\\{G({\mathbb{a}}_{1}),\dots,G({\mathbb{a}}_{n})\\}$ and we will call
$G({\mathbb{a}}_{i})$ the Gale transform of ${\mathbb{a}}_{i}$. There is an
edge between the vertices ${\mathbb{a}}_{i}$ and ${\mathbb{a}}_{j}$ if and
only if ${\mathbb{a}}_{i}$, ${\mathbb{a}}_{j}$ are coparallel, that means
$G({\mathbb{a}}_{i}),G({\mathbb{a}}_{j})$ are parallel i.e. there exists a
nonzero $\lambda\in{\mathbb{Q}}$ such that $G({\mathbb{a}}_{i})=\lambda
G({\mathbb{a}}_{j})$.

The vector ${\bf a}_{i}$ is called free if $G({\mathbb{a}}_{i})$ is equal to
the zero vector, which means that $i$ is not contained in the support of any
element in $\operatorname{Ker}_{\mathbb{Z}}(A)$. It follows from the
definition that the free vectors of $A$ form one bouquet, which we call the
free bouquet of $G_{A}$. If the configuration of weights ${\bf a}_{1},{\bf
a}_{2},\cdots,{\bf a}_{n}$ has no free vectors then it is called non-
pyramidal. If the configuration has $k$ free vectors is called $k$-pyramidal.
The non-free bouquets are of two types: mixed and non-mixed. A non-free
bouquet is mixed if contains an edge $\\{{\mathbb{a}}_{i},{\mathbb{a}}_{j}\\}$
such that $G({\mathbb{a}}_{i})=\lambda G({\mathbb{a}}_{j})$ for some
$\lambda<0$, and is non-mixed if it is either an isolated vertex or for all of
its edges $\\{{\mathbb{a}}_{i},{\mathbb{a}}_{j}\\}$ we have
$G({\mathbb{a}}_{i})=\lambda G({\mathbb{a}}_{j})$ with $\lambda>0$, see [24,
Lemma 1.2].

Let $B_{1},\ldots,B_{s}$ be the bouquets of $A$. Then we may rearrange the
column vectors in such a way that there exist positive integers
$i_{1},i_{2},\ldots,i_{s}$ such that the first $i_{1}$ column vectors
${\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{i_{1}}$ belong to the first bouquet
$B_{1}$, the next $i_{2}$ column vectors belong to the second bouquet $B_{2}$,
and so on the last $i_{s}$ vectors belong to the $s$-th bouquet $B_{s}$, thus
in particular $i_{1}+\cdots+i_{s}=n$.

Next, we recall from [24], the definition of the _bouquet-index-encoding
vector_ ${\mathbb{c}}_{B}$. If the bouquet $B$ is free then we set
${\mathbb{c}}_{B}\in{\mathbb{Z}}^{n}$ to be any nonzero vector such that
$\operatorname{supp}({\mathbb{c}}_{B})=\\{i|{\mathbb{a}}_{i}\in B\\}$ and with
the property that the first nonzero component is positive. For a non-free
bouquet $B$ of $A$, consider the Gale transforms of the elements in $B$. All
the elements are nonzero and pairwise linearly dependent, therefore there
exists at least one nonzero component $j$ in all of them. Let
$g_{j}=\gcd(G({\bf a}_{i})_{j}|\ {\bf a}_{i}\in B)$ and fix the smallest
integer $\ell$ such that ${\bf a}_{\ell}\in B$. Let ${\bf c}_{B}$ be the
vector in ${\mathbb{Z}}^{n}$ whose $i$-th component is $0$ if ${\bf
a}_{i}\notin B$, and is $\varepsilon_{\ell j}G({\bf a}_{i})_{j}/g_{j}$ if
${\bf a}_{i}\in B$, where $G({\mathbb{a}}_{i})_{j}$ represents the $j$-th
component of the vector $G({\mathbb{a}}_{i})$ and $\varepsilon_{\ell j}$ the
sign of the integer $G({\bf a}_{\ell})_{j}$. Note that the choice of $\ell$
implies that the first nonzero component of ${\bf c}_{B}$ is positive.
Division by $g_{j}$ makes the gcd of the components of ${\bf c}_{B}$ to be
one. Since each ${\bf a}_{i}$ belongs to exactly one bouquet the supports of
the vectors ${\bf c}_{B_{i}}$ are pairwise disjoint. In addition,
$\cup_{i}\operatorname{supp}({\mathbb{c}}_{B_{i}})=[n]$, and taking into
account that the first $i_{1}$ vectors are in $B_{1}$, the next $i_{2}$ in
$B_{2}$, and so forth, then ${\mathbb{c}}_{B_{1}}$ has only the first $i_{1}$
components nonzero, ${\mathbb{c}}_{B_{2}}$ has only the following $i_{2}$
components nonzero, and so forth. Therefore, we may label the nonzero
components of ${\mathbb{c}}_{B_{1}},\ldots,{\mathbb{c}}_{B_{s}}$ as follows:
the nonzero components of ${\mathbb{c}}_{B_{q}}$ are
$c_{q1},\ldots,c_{qi_{q}}$, for all $1\leq q\leq s$.

The structure of a non-free bouquet $B$ of $A$, that is mixed or non-mixed,
can be detected from the vector ${\mathbb{c}}_{B}$. If $B_{i}$ is a non-free
bouquet of $A$, then $B_{i}$ is a mixed bouquet if and only if the vector
${\bf c}_{B_{i}}$ has a negative and a positive component, and $B_{i}$ is non-
mixed if and only if the vector ${\mathbb{c}}_{B_{i}}$ has all nonzero
components positive, see [24, Lemma 1.6].

For each bouquet $B$ we define the following vector:

${\mathbb{a}}_{B}=\sum_{j=1}^{n}(c_{B})_{j}{\mathbb{a}}_{j}\in{\mathbb{Z}}^{m},$

where $(c_{B})_{j}$ is the $j$-th component of the vector ${\mathbb{c}}_{B}$.
Note that the ${\bf a}_{j}$’s that contribute in fact to the vector ${\bf
a}_{B}$ are those belonging to the bouquet $B$. The matrix $A_{B}$ whose
column vectors are the vectors
${\mathbb{a}}_{B_{1}},{\mathbb{a}}_{B_{2}},\ldots,{\mathbb{a}}_{B_{s}}$,
corresponding to the bouquets of $A$, is called the bouquet matrix of $A$.

Finally, we recall (see [24, Theorem 1.9]) that the matrix
$A_{B}\in{\mathbb{Z}}^{m\times s}$ has its kernel
$\operatorname{Ker}_{{\mathbb{Z}}}(A_{B})$ isomorphic to
$\operatorname{Ker}_{{\mathbb{Z}}}(A)$ via the isomorphism
$D:\operatorname{Ker}_{{\mathbb{Z}}}(A_{B})\mapsto\operatorname{Ker}_{{\mathbb{Z}}}(A)$
given by

$D(u_{1},\ldots,u_{q})=(c_{11}u_{1},\ldots,c_{1i_{1}}u_{1},c_{21}u_{2},\ldots,c_{2i_{2}}u_{2},\ldots,c_{s1}u_{s},\ldots,c_{si_{s}}u_{s}).$

Note that the vector $D({\bf u})\in\operatorname{Ker}_{{\mathbb{Z}}}(A)$ has
$n$ components, which we will denote by $u_{jt}$ in the rest of the paper,
where $1\leq j\leq s$, $1\leq t\leq i_{j}$.

The following theorem describes the connection between the toric ideals of $A$
and $A_{B}$. By $\mathcal{C}(A)$ we denote the circuits of $A$, which consists
of all nonzero vectors ${\bf{u}}\in\operatorname{Ker}_{{\mathbb{Z}}}(A)$ with
minimal support and the gcd of their components equal to one. While by
$\operatorname{Gr}(A)$ we denote the Graver basis of $A$, which consists of
all nonzero vectors ${\bf{u}}\in\operatorname{Ker}_{{\mathbb{Z}}}(A)$ with no
proper conformal decomposition, see [26, Section 4]. We say that ${\bf u}={\bf
v}+_{c}{\bf w}$ is a conformal decomposition of the vector ${\bf u}$ in two
vectors v, w, if ${\bf u}={\bf v}+{\bf w}$ and ${{\bf u}}^{+}={{\bf
v}}^{+}+{{\bf w}}^{+},{{\bf u}}^{-}={{\bf v}}^{-}+{{\bf w}}^{-}$. We call the
decomposition proper if both ${\bf{v}}$ and ${\bf{w}}$ are nonzero.

######  Theorem 2.1 ([24, Theorems 1.9 and 1.11]).

Let $A=[{\bf a}_{1},\dots,{\bf a}_{n}]\in{\mathbb{Z}}^{d\times n}$ and its
bouquet matrix $A_{B}=[{\mathbb{a}}_{B_{1}},\dots,{\mathbb{a}}_{B_{s}}]$.
There is a bijective correspondence between the elements of
$\operatorname{Ker}_{{\mathbb{Z}}}(A_{B})$ $(\operatorname{Gr}(A_{B})$ and
$\mathcal{C}(A_{B})$, respectively) and the elements of
$\operatorname{Ker}_{{\mathbb{Z}}}(A)$ $(\operatorname{Gr}(A)$ and
$\mathcal{C}(A)$, respectively). More precisely, this correspondence is
defined as follows: if ${\bf
u}=(u_{1},\ldots,u_{s})\in\operatorname{Ker}_{{\mathbb{Z}}}(A_{B})$, then
$D({\bf u})={\bf c}_{B_{1}}u_{1}+\cdots+{\bf
c}_{B_{s}}u_{s}\in\operatorname{Ker}_{{\mathbb{Z}}}(A)$.

######  Lemma 2.2.

Let $A=\\{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{n}\\}$ be a configuration and
$A_{B}=\\{{\mathbb{a}}_{B_{1}},\ldots,{\mathbb{a}}_{B_{s}}\\}$ be its bouquet
configuration. Then $A$ is non-pyramidal if and only if $A_{B}$ is non-
pyramidal.

###### Proof.

Suppose that $A$ is a non-pyramidal configuration and $A_{B}$ is pyramidal.
Then there exists an ${{\bf a}_{B}}_{j}$ which is a free vector, thus $G({{\bf
a}_{B}}_{j})={\bf 0}$. From the definition of the Gale transform this means
that every element ${\bf u}$ in the $\operatorname{Ker}_{{\mathbb{Z}}}(A_{B})$
has the $j$-th component equal to $0$. Then, according to Theorem 2.1, all
elements $D({\bf u})$ in the $\operatorname{Ker}_{{\mathbb{Z}}}(A)$ have their
components $u_{j1},\cdots,u_{ji_{j}}$ corresponding to the $B_{j}$ bouquet
zero, since $u_{jt}=c_{jt}u_{j}=0$. Therefore all the Gale transforms of the
vectors of the bouquet $B_{j}$ are zero and thus the vectors are free,
contradicting the fact that $A$ is non-pyramidal configuration.

For the other direction, suppose that $A_{B}$ is a non-pyramidal configuration
and $A$ is a pyramidal. Then there exists a free vector in $A$, thus its Gale
transform is zero. Suppose that this free vector is the $l$-th one in the
bouquet $B_{j}$. Then all elements $D({\bf u})$ in the
$\operatorname{Ker}_{{\mathbb{Z}}}(A)$ have their $jl$-th component equal to
$0$. According to Theorem 2.1, the $jl$-th component is $c_{jl}u_{j}$. But
$c_{jl}\not=0$ thus $u_{j}=0$, for all ${\bf
u}\in\operatorname{Ker}_{{\mathbb{Z}}}(A_{B})$. Therefore, the Gale transform
of the vector ${{\bf a}_{B}}_{j}$ is zero and thus the vector ${{\bf
a}_{B}}_{j}$ is free, contradicting the fact that $A_{B}$ is a non-pyramidal
configuration. ∎

##  3\. Generalized Lawrence matrices and Classification results

Next we recall the definition of a generalized Lawrence matrix from [24,
Section 2]. Let
$\\{{\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}\\}\subset{\mathbb{Z}}^{d}$
be a configuration and
${\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s}$ be any set of
primitive vectors, a primitive vector is an integer vector such that the gcd
of all its components is $1$, with
${\mathbb{c}}^{\prime}_{i}\in{\mathbb{Z}}^{m_{i}}$ for some $m_{i}\geq 1$,
each having full support and a positive first component. We denote a
generalized Lawrence matrix by
$\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$
and is defined as follows: for each $i=1,\dots,s$, let
${\mathbb{c}}^{\prime}_{i}=(c^{\prime}_{i1},\ldots,c^{\prime}_{im_{i}})\in{\mathbb{Z}}^{m_{i}}$
and define

$C_{i}=\left(\begin{array}[]{ccccc}-c^{\prime}_{i2}&c^{\prime}_{i1}&0&\ldots&0\\\
-c^{\prime}_{i3}&0&c^{\prime}_{i1}&\ldots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
-c^{\prime}_{im_{i}}&0&0&\ldots&c^{\prime}_{i1}\end{array}\right)\in{\mathbb{Z}}^{(m_{i}-1)\times
m_{i}}.$

Primitivity of each ${\mathbb{c}}^{\prime}_{i}$ implies that there exist
integers $\lambda_{i1},\ldots,\lambda_{im_{i}}$ such that
$1=\lambda_{i1}c^{\prime}_{i1}+\cdots+\lambda_{im_{i}}c^{\prime}_{im_{i}}$.
Fix a choice of $\lambda_{i1},\ldots,\lambda_{im_{i}}$, and define the
matrices $A_{i}=[\lambda_{i1}{\bf a}^{\prime}_{i},\ldots,\lambda_{im_{i}}{\bf
a}^{\prime}_{i}]\in{\mathbb{Z}}^{d\times m_{i}}$. The generalized Lawrence
matrix is then the following block matrix:

$\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})=\
\left(\begin{array}[]{cccc}A_{1}&\ A_{2}&\cdots&\ A_{s}\\\ C_{1}&\
0&\cdots&0\\\ 0&\ C_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&\
0&\cdots&\ C_{s}\end{array}\right)\in{\mathbb{Z}}^{p\times n},$

where $p=d+(m_{1}-1)+\cdots+(m_{s}-1)$ and $n=m_{1}+\cdots+m_{s}$, see 3.6 for
an example of a generalized Lawrence matrix. Note that the definition of a
generalized Lawrence matrix depends also on the choice of $\lambda_{ij}$, and
thus there are infinitely many choices for
$\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$.
However, all of them have the same kernel and thus define the same toric
variety, see [24, Section 2].

######  Theorem 3.1 ([24, Theorem 2.1]).

Let
$\\{{\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}\\}\subset{\mathbb{Z}}^{d}$
be an arbitrary set of vectors. Let
${\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s}$ be any set of
primitive vectors, with ${\mathbb{c}}^{\prime}_{i}\in{\mathbb{Z}}^{m_{i}}$ for
some $m_{i}\geq 1$, each having full support and a positive first component.
Then, any generalized Lawrence matrix
$A=\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})\in{\mathbb{Z}}^{p\times
n}$ has a subbouquet decomposition, $B_{1},\ldots,B_{s}$, such that the
$i^{th}$ subbouquet is encoded by the following vectors:
${\mathbb{a}}_{B_{i}}=({\mathbb{a}}^{\prime}_{i},{\bf 0},\ldots,{\bf
0})\in{\mathbb{Z}}^{p}$ and ${\mathbb{c}}_{B_{i}}=({\bf
0},\ldots,{\mathbb{c}}^{\prime}_{i},\ldots,{\bf 0})\in{\mathbb{Z}}^{q}$, where
$n=m_{1}+\cdots+m_{s}$, $p=d+q-s$ and the support of ${\mathbb{c}}_{B_{i}}$ is
precisely the $i^{th}$ block of ${\mathbb{Z}}^{n}$ of size $m_{i}$.

######  Lemma 3.2.

Let
$A=\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$
be such that $\sum_{j}c^{\prime}_{ij}=0$ for all $i=1,\ldots,s$. Then $X_{A}$
is a projective toric variety.

###### Proof.

Adding up the rows of $C_{i}$ we obtain the row vector
$(-c^{\prime}_{i2}-\cdots-c^{\prime}_{im_{i}},c^{\prime}_{i1},\ldots,c^{\prime}_{i1})$
which is equal by the hypothesis with
$(c^{\prime}_{i1},c^{\prime}_{i1},\ldots,c^{\prime}_{i1})$, and since
$c^{\prime}_{i1}\neq 0$ we conclude that ${\bf
1}_{i}=(1,\ldots,1)\in{\mathbb{Z}}^{m_{i}}$ lies in the rowspan of $C_{i}$.
Since this can be done for every $i$ by adding the corresponding row vectors
$({\bf 0},\ldots,{\bf 1}_{i},\ldots,{\bf 0})\in{\mathbb{Z}}^{n}$ of $A$ we
obtain that $(1,\ldots,1)\in{\mathbb{Z}}^{n}$ lies in the rowspan of $A$. ∎

Therefore the projectiveness of $X_{A}$, when
$A=\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$,
does not depend on the vectors ${\mathbb{a}}^{\prime}_{i}$, in other words on
the bouquet ideal, but only on the property of the bouquet-index-encoding
vectors that $\sum_{j}c^{\prime}_{ij}=0$ for all $i=1,\ldots,s$. For example
in 3.6 the bouquet ideal is the ideal of an affine monomial curve.

The next theorem is a reformulation of [5, Theorem 4.4] since the vectors that
their Gale transforms are parallel form a bouquet.

######  Theorem 3.3.

Let $A=[{\mathbb{a}}_{1}\cdots{\mathbb{a}}_{n}]$ be a non pyramidal
configuration. Then $X_{A}$ is self-dual if and only if
$\sum_{{}_{{\mathbb{a}}_{i}\in B}}G({\mathbb{a}}_{i})={\bf 0}$ for any bouquet
$B$ of $A$.

######  Theorem 3.4.

Let $A=[{\mathbb{a}}_{1}\cdots{\mathbb{a}}_{n}]$ be a non pyramidal
configuration. Then $X_{A}$ is self-dual if and only if
$\sum_{i}(c_{B})_{i}=0$ for any bouquet $B$ of $A$.

###### Proof.

The configuration is non pyramidal therefore all bouquets are non-free. We
recall the definition of the vector ${\bf c}_{B}$. Let $j$ be such that all
$j$-components of the Gale transforms of the elements in $B$ are nonzero. Let
$g_{j}=\gcd(G({\bf a}_{i})_{j}|\ {\bf a}_{i}\in B)$ and fix the smallest
integer $\ell$ such that ${\bf a}_{\ell}\in B$. Then
$(c_{B})_{i}=\varepsilon_{\ell j}G({\mathbb{a}}_{i})_{j}/g_{j}$, where
$\varepsilon_{\ell j}$ represents the sign of the integer $G({\bf
a}_{\ell})_{j}$. From Theorem 3.3 we have $\sum_{{}_{{\mathbb{a}}_{i}\in
B}}G({\mathbb{a}}_{i})={\bf 0}$ for any bouquet $B$ of $A$. Then the sum of
their $j$-components is $0$, thus $\sum_{i}G({\mathbb{a}}_{i})_{j}=0$
therefore $\sum_{i}\varepsilon_{\ell j}G({\mathbb{a}}_{i})_{j}/g_{j}=0$. We
conclude that $\sum_{i}(c_{B})_{i}=0$ for any bouquet $B$ of $A$. ∎

######  Theorem 3.5.

Let
$A=\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$
be a generalized Lawrence matrix with
${\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}$ a non-pyramidal
configuration. Then $X_{A}$ is a self-dual projective toric variety if and
only if $\sum_{j}c^{\prime}_{ij}=0$ for all $i=1,\ldots,s$.

###### Proof.

From [24, Theorem 2.1, Corollary 2.3] we know that $A$ has exactly $s$
bouquets and the encoding bouquet vectors of $A$ are
${\mathbb{a}}_{B_{i}}=({\mathbb{a}}^{\prime}_{i},{\bf 0},\ldots,{\bf
0})\in{\mathbb{Z}}^{p}$ and ${\mathbb{c}}_{B_{i}}=({\bf
0},\ldots,{\mathbb{c}}^{\prime}_{i},\ldots,{\bf 0})\in{\mathbb{Z}}^{n}$ for
$i=1,\ldots,s$. Applying Lemma 2.2 for $A$ we obtain that $A$ is a non-
pyramidal configuration. The conclusion follows now from Theorem 3.4. ∎

Theorem 3.5 gives a way of producing examples of non-pyramidal self-dual
projective toric varieties. Take any matrix with columns
${\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}$ to be a non-
pyramidal configuration, then take a set of $s$ primitive vectors
${\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s}$ with the sum of
their components zero. Then the toric variety defined by the matrix
$\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$
is a non-pyramidal self-dual projective toric variety.

######  Example 3.6.

Let $A$ be any configuration, for example the one given by the columns of the
matrix $A=(4\ 5\ 6\ 7)$. Take a primitive vector for each column such that the
sum of the components for each primitive vector is zero, say
$c^{\prime}_{1}=(1,-1),c^{\prime}_{2}=(7,-4,-3),c^{\prime}_{3}=(2023,1,-2024),c^{\prime}_{4}=(1,3,-2,-2)$.
Computing any integer solution for each
$1=\lambda_{i1}c^{\prime}_{i1}+\cdots+\lambda_{im_{i}}c^{\prime}_{im_{i}}$,
for example $1=1\cdot 1+0\cdot(-1)$, $1=1\cdot 7+0\cdot(-4)+2\cdot(-3)$,
$1=0\cdot 2023+1\cdot 1+0\cdot(-2024)$ and $1=0\cdot 1+1\cdot
3+1\cdot(-2)+0\cdot(-2)$ we get the generalized Lawrence matrix
$C\in{\mathbb{Z}}^{9\times 12}$:

$\tiny{\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{4}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{4})=\
\left(\begin{array}[]{cccccccccccc}4&0&5&0&10&0&6&0&0&7&7&0\\\
1&1&0&0&0&0&0&0&0&0&0&0\\\ 0&0&4&7&0&0&0&0&0&0&0&0\\\
0&0&3&0&7&0&0&0&0&0&0&0\\\ 0&0&0&0&0&-1&2023&0&0&0&0&0\\\
0&0&0&0&0&2024&0&2023&0&0&0&0\\\ 0&0&0&0&0&0&0&0&-3&1&0&0\\\
0&0&0&0&0&0&0&0&2&0&1&0\\\ 0&0&0&0&0&0&0&0&2&0&0&1\end{array}\right).}$

Theorem 3.5 proves that $X_{C}$ is a self-dual projective toric variety.

Next Theorem 3.7 actually says that all non-pyramidal configurations are taken
in the way described above.

######  Theorem 3.7.

Let $A$ be a non-pyramidal configuration such that $X_{A}$ is self-dual. Then
there exists a generalized Lawrence matrix
$A^{\prime}=\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$
such that $X_{A}=X_{A^{\prime}}$ after a possible renumeration of the columns,
where
${\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}\in{\mathbb{Z}}^{d}$
is a non-pyramidal configuration and $\sum_{j}c^{\prime}_{ij}=0$ for all
$i=1,\ldots,s$.

###### Proof.

For the integer matrix $A$ we know from [24, Corollary 2.3] that there exists
a generalized Lawrence matrix
$A^{\prime}=\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$
such that $I_{A}=I_{A^{\prime}}$, up to permutation of column indices. Then
$A^{\prime}$ is also a non-pyramidal configuration, therefore by Lemma 2.2 and
Theorem 3.1 the configuration
${\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}\in{\mathbb{Z}}^{d}$
is non-pyramidal. The last condition that $\sum_{j}c^{\prime}_{ij}=0$ for all
$i=1,\ldots,s$ follows from Theorem 3.5. ∎

For the next result we need one notation: the vector
$\varepsilon_{i}\in{\mathbb{Z}}^{d}$, $1\leq i\leq d$, represents the integer
vector with the only nonzero component equal to $1$ and on the $i$-th
position.

######  Theorem 3.8.

Consider the generalized Lawrence matrix

$C=\operatorname{GLM}(\varepsilon_{1},{\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{0},{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$

with $n$ columns, ${\mathbb{c}}^{\prime}_{0}\in{\mathbb{Z}}^{k}$,
$c^{\prime}_{01}=1$, $k\geq 1$, ${\mathbb{a}}^{\prime}_{i}=(0,*,\ldots,*)$ and
$\sum_{j}c^{\prime}_{ij}=0$ for all $i=1,\ldots,s$. If the configuration
$\\{{\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}\\}\subset{\mathbb{Z}}^{d}$
is non-pyramidal, then for any $n+k$ multiset configuration $A$ with the
ground set the set of columns of $C$ we have that $X_{A}$ is a self-dual
projective toric variety.

###### Proof.

First we note that, by the proof of Lemma 3.2, the vector
$(0,\ldots,0,1,\ldots,1)\in{\mathbb{Z}}^{n}$ lies in the rowspan of $C$, where
the first $k$ components are zero and the rest are one. The matrix

$\left(\begin{array}[]{c}A_{0}\\\
C_{0}\end{array}\right)=\left(\begin{array}[]{ccccc}\lambda_{01}&\lambda_{02}&\lambda_{03}&\ldots&\lambda_{0k}\\\
-c^{\prime}_{02}&c^{\prime}_{01}&0&\ldots&0\\\
-c^{\prime}_{03}&0&c^{\prime}_{01}&\ldots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
-c^{\prime}_{0k}&0&0&\ldots&c^{\prime}_{01}\end{array}\right)=\left(\begin{array}[]{ccccc}1&0&0&\ldots&0\\\
-c^{\prime}_{02}&1&0&\ldots&0\\\ -c^{\prime}_{03}&0&1&\ldots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
-c^{\prime}_{0k}&0&0&\ldots&1\end{array}\right)\in{\mathbb{Z}}^{k\times k},$

since we can choose as the solution to
$1=\lambda_{01}c^{\prime}_{01}+\cdots+\lambda_{0k}c^{\prime}_{0k}$ with
$c^{\prime}_{01}=1$ the $\lambda_{01}=1$ and $\lambda_{0j}=0$, for all $2\leq
j\leq k$. Therefore in the row span of the matrix
$\left(\begin{array}[]{c}A_{0}\\\ C_{0}\end{array}\right)$ belongs every
vector of ${\mathbb{Z}}^{k}$, thus also the $(1,\ldots,1)\in{\mathbb{Z}}^{k}$.
Note that the first row of $C$ and the $d+1$-th up to $(d+k)$-th rows have the
last $n-k$ components equal to 0. Concluding, we see that
$(1,\ldots,1,0,\ldots,0)\in{\mathbb{Z}}^{n}$ lies in the row span of $C$,
where the first $k$ components are one and the rest zero. Combining the two
results up to this point we get that
$(1,\ldots,1,1,\ldots,1)\in{\mathbb{Z}}^{n}$ belongs to the row span of $C$
and so the toric variety $X_{C}$ is projective. $A$ is a multiset
configuration with ground set the set of columns of $C$, therefore the toric
variety $X_{A}$ is also projective.

Theorem 3.8 of [5] states that $X_{A}$ is self dual if and only if (1) the
configuration $C$ is $k$-pyramidal and (2) there exists a splitting of the
lattice of characters ${\mathbb{Z}}^{p}$ as $S_{1}\times S_{2}$ such that,
after reordering of the elements of $C$, it holds that
$C={\mathcal{C}}_{1}\cup{\mathcal{C}}_{2}$, where ${\mathcal{C}}_{1}$ is a
basis of $\chi(S_{1})$ and ${\mathcal{C}}_{2}\subset\chi(S_{2})$ is a non-
pyramidal configuration and the toric variety $X_{{\mathcal{C}}_{2}}$ is self-
dual.

We denote the vector columns of the matrix $C\in{\mathbb{Z}}^{p\times n}$ by
${\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{n}$. The vectors in
${\mathcal{C}}_{1}=\\{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{k}\\}$ are free
vectors since there exist vectors ${\mathbb{u}}_{i}\in{\mathbb{Q}}^{p}$ with
$i=1,\ldots,k$ such that

$({\mathbb{u}}_{i}\cdot{\mathbb{a}}_{1},\ldots,{\mathbb{u}}_{i}\cdot{\mathbb{a}}_{n})=\varepsilon_{i}\in{\mathbb{Z}}^{p}.$

Indeed, the vector ${\mathbb{u}}_{1}=\varepsilon_{1}$ and the vectors
${\mathbb{u}}_{i}=c^{\prime}_{0i}\varepsilon_{1}+\varepsilon_{d+i-1}$ for all
$i=2,\ldots,k$ satisfy the above conditions, since

$C=\left(\begin{array}[]{ccccc}\epsilon_{1}&0&\ 0&\cdots&\ 0\\\ 0&A_{1}&\
A_{2}&\cdots&\ A_{s}\\\ C_{0}&0&\ 0&\cdots&\ 0\\\ 0&C_{1}&\ 0&\cdots&\ 0\\\
0&0&\ C_{2}&\cdots&\ 0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&\
0&\cdots&\ C_{s}\end{array}\right)\in{\mathbb{Z}}^{p\times n},$

where the vector $\varepsilon_{1}\in{\mathbb{Z}}^{k}$. Thus
${\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{k}$ are free vectors, therefore the
condition (1) of [5, Theorem 3.8] is satisfied. Let

$\chi(S_{1})=\\{(v_{1},0,0,\cdots,0,v_{d+1},\cdots,v_{d+k},0,\cdots,0)|v_{i}\in{\mathbb{Z}}\\},$
$\chi(S_{2})=\\{(0,v_{2},v_{3},\cdots,v_{d},0,\cdots,0,v_{d+k+1},\cdots,v_{p})|v_{i}\in{\mathbb{Z}}\\},$

then ${\mathbb{Z}}^{p}=\chi(S_{1})\times\chi(S_{2})$. So ${\mathcal{C}}_{1}$
is a basis of $\chi(S_{1})$. Let
${\mathcal{C}}_{2}=\\{{\mathbb{a}}_{k+1},\ldots,{\mathbb{a}}_{n}\\}$ be the
remaining column vectors of $C$. Denote also by ${\mathcal{C}}_{2}$ the matrix
with columns ${\mathbb{a}}_{k+1},\ldots,{\mathbb{a}}_{n}$. The two matrices
${\mathcal{C}}_{2}$ and
$\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$
have the same kernel, since the two matrices differ by some rows of zeroes.
Thus they define the same toric ideal and variety.

The configuration
$\\{{\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}\\}$ is non-
pyramidal. It follows from Lemma 2.2 and Theorem 3.1 that the columns of the
generalized Lawrence matrix

$\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$

form a non-pyramidal configuration. Thus ${\mathcal{C}}_{2}\subset\chi(S_{2})$
is a non-pyramidal configuration. The toric variety $X_{{\mathcal{C}}_{2}}$ is
self-dual by Theorem 3.5, since $\sum_{j}c^{\prime}_{ij}=0$ for all
$i=1,\ldots,s$.

Both conditions of [5, Theorem 3.8] are satisfied thus $X_{A}$ is a self-dual
projective toric variety. ∎

Theorem 3.8 gives a way of producing examples of pyramidal or non-pyramidal
self-dual projective toric varieties. Take any matrix with columns
${\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}$, with
${\mathbb{a}}^{\prime}_{i}=(0,*,\ldots,*)$, to be a non-pyramidal
configuration, then take a set of $s$ primitive vectors
${\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s}$ with the sum of
their components zero. Then create the $p\times n$-matrix
$C=\operatorname{GLM}(\varepsilon_{1},{\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{0},{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$
with ${\mathbb{c}}^{\prime}_{0}\in{\mathbb{Z}}^{k}$, $c^{\prime}_{01}=1$ and
$k\geq 1$. Take any $n+k$ multiset configuration $A$ with the ground set the
set of columns of $C$, then $X_{A}$ is a self-dual projective toric variety.
Theorem 3.11 determines under what conditions on the multiset $A$ the
resulting self-dual toric variety will be pyramidal or not. Next Theorem 3.9
says that all self-dual projective toric varieties are produced in this way.

######  Theorem 3.9.

Let $A$ be a multiset configuration with the ground set $C$ such that $X_{A}$
is self-dual. Then there exists a generalized Lawrence matrix

$C^{\prime}=\operatorname{GLM}(\varepsilon_{1},{\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{0},{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$

and a multiset configuration $A^{\prime}$ with ground set the set of columns
of $C^{\prime}$, such that $X_{A}=X_{A^{\prime}}$ after a possible
renumeration of the columns, where $k=|A|-|C|$,
${\mathbb{c}}^{\prime}_{0}\in{\mathbb{Z}}^{k}$, $c^{\prime}_{01}=1$,
${\mathbb{a}}^{\prime}_{i}=(0,*,\ldots,*)\in{\mathbb{Z}}^{d}$,
${\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}$ a non-pyramidal
configuration and $\sum_{j}c^{\prime}_{ij}=0$ for all $i=1,\ldots,s$.

###### Proof.

Let $A$ be the multiset configuration
$\\{\underbrace{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{1}}_{k_{1}+1},\ldots,\underbrace{{\mathbb{a}}_{n},\ldots,{\mathbb{a}}_{n}}_{k_{n}+1}\\}$
with the ground set $C=\\{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{n}\\}$ such
that $X_{A}$ is self-dual and $k=\sum_{i=1}^{n}k_{i}$. Then from [5, Theorem
3.8] the configuration $C$ is $k$-pyramidal and there exists a splitting of
the lattice of characters ${\mathbb{Z}}^{p}$ as $S_{1}\times S_{2}$ such that,
after reordering of the elements of $C$, it holds that
$C={\mathcal{C}}_{1}\cup{\mathcal{C}}_{2}$, where ${\mathcal{C}}_{1}$ is a
basis of $\chi(S_{1})$ and ${\mathcal{C}}_{2}\subset\chi(S_{2})$ is a non-
pyramidal configuration and the toric variety $X_{{\mathcal{C}}_{2}}$ is self-
dual. $\chi(S_{1})\simeq{\mathbb{Z}}^{k}$ and we can suppose that
${\mathcal{C}}_{1}=\\{\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{k}\\}\subset\chi(S_{1})=\\{(v_{1},v_{2},\cdots,v_{k},0,\cdots,0)|v_{i}\in{\mathbb{Z}}\\}$.
Let
$\chi(S_{2})=\\{(0,\cdots,0,v_{k+1},v_{k+2},\cdots,v_{p})|v_{i}\in{\mathbb{Z}}\\}$.
For the integer matrix ${\mathcal{C}}_{2}$ we know from [24, Corollary 2.3]
that there exists a generalized Lawrence matrix
${\mathcal{C}}_{2}^{\prime}=\operatorname{GLM}({\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$
such that $I_{{\mathcal{C}}_{2}}=I_{{\mathcal{C}}^{\prime}_{2}}$, up to a
permutation of column indices. Therefore ${\mathcal{C}}_{2}$ and
${\mathcal{C}}^{\prime}_{2}$ have the same integer kernel, up to a permutation
of the components, as well as the matrices
$C=\left(\begin{array}[]{cc}I_{k}&{\bf 0}\\\ {\bf
0}&{\mathcal{C}}_{2}\end{array}\right)$ and
$\left(\begin{array}[]{cc}I_{k}&{\bf 0}\\\ {\bf
0}&{\mathcal{C}}_{2}^{\prime}\end{array}\right)$. The second matrix after the
row operations $R_{i}\leftrightarrow R_{d+i}$, $2\leq i\leq k$, and then
$R_{i}\rightarrow R_{i}-c^{\prime}_{0i}R_{1}$, $2\leq i\leq k$, which do not
change the kernel can be brought to the form
$C^{\prime}=\operatorname{GLM}(\varepsilon_{1},{\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}|{\mathbb{c}}^{\prime}_{0},{\mathbb{c}}^{\prime}_{1},\ldots,{\mathbb{c}}^{\prime}_{s})$.
By Lemma 2.2 the configuration
${\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{s}$ is non-pyramidal
and by Theorem 3.5 the $\sum_{j}c^{\prime}_{ij}=0$ for all $i=1,\ldots,s$. Let
$\sigma$ be the permutation such that
$\operatorname{Ker}_{{\mathbb{Z}}}(C)=\operatorname{Ker}_{{\mathbb{Z}}}(\sigma(C^{\prime}))$.
Let ${\mathbb{b}}_{1},\ldots,{\mathbb{b}}_{n}$ be the columns of $C^{\prime}$
and $A^{\prime}$ be the multiset configuration
$\\{\underbrace{{\mathbb{b}}_{\sigma(1)},\ldots,{\mathbb{b}}_{\sigma(1)}}_{k_{1}+1},\ldots,\underbrace{{\mathbb{b}}_{\sigma(n)},\ldots,{\mathbb{b}}_{\sigma(n)}}_{k_{n}+1}\\}$
then
$\operatorname{Ker}_{{\mathbb{Z}}}(C)=\operatorname{Ker}_{{\mathbb{Z}}}(\sigma(C^{\prime}))$
implies
$\operatorname{Ker}_{{\mathbb{Z}}}(A)=\operatorname{Ker}_{{\mathbb{Z}}}(A^{\prime})$.
Thus $X_{A}=X_{A^{\prime}}$. ∎

######  Example 3.10.

Let $A_{1}$ be a simple 1-pyramidal configuration, for example the one given
by the columns of the matrix $\left(\begin{array}[]{ccccc}1&0&0&0&0\\\
0&4&5&6&7\end{array}\right)$.

Take a primitive vector for each column such that the sum of the component for
each primitive vector is zero, except possible for the first one say
$c^{\prime}_{0}=(1,3,5,7),c^{\prime}_{1}=(1,-1),c^{\prime}_{2}=(7,-4,-3),c^{\prime}_{3}=(2023,1,-2024),c^{\prime}_{4}=(1,3,-2,-2)$.
Computing any integer solution for each
$1=\lambda_{i1}c^{\prime}_{i1}+\cdots+\lambda_{im_{i}}c^{\prime}_{im_{i}}$
(for $i=0$ a solution is $1=1\cdot 1+0\cdot 3+0\cdot 5+0\cdot 7$) we get a
generalized Lawrence matrix
$\operatorname{GLM}(\varepsilon_{1},{\mathbb{a}}^{\prime}_{1},\ldots,{\mathbb{a}}^{\prime}_{4}|{\mathbb{c}}^{\prime}_{0},\ldots,{\mathbb{c}}^{\prime}_{4})$
as follows

$C_{1}=\tiny{\left(\begin{array}[]{cccccccccccccccc}1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&4&0&5&0&10&0&6&0&0&7&7&0\\\ -3&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
-5&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ -7&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&1&1&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&4&7&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&3&0&7&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&-1&2023&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&2024&0&2023&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&-3&1&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&2&0&1&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&2&0&0&1\end{array}\right).}$

Let $E$ be any 16+4 multiset configuration with the ground set the set of
columns of $C_{1}$. Theorem 3.8 proves that $X_{E}$ is a self-dual projective
toric variety. Theorem 3.9 says that all multiset configurations such that
$X_{E}$ is a self-dual projective toric variety are taken in this way. Note
that if we repeat once all first four columns the corresponding variety is
non-pyramidal, while in all other cases is s-pyramidal, where s is less than
or equal to four and depends on the choices of the four vectors that we
repeat, see the examples 4.8, 4.9. This is the subject of the next Theorem.

######  Theorem 3.11.

Let

$A=\\{\underbrace{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{1}}_{k_{1}+1},\ldots,\underbrace{{\mathbb{a}}_{n},\ldots,{\mathbb{a}}_{n}}_{k_{n}+1}\\}$

a multiset configuration with $k_{1},\ldots,k_{n}$ nonnegative integers with
the ground set
$C=\\{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{n}\\}\subset{\mathbb{Z}}^{d}$,
${\mathbb{a}}_{i}\neq{\mathbb{a}}_{j}$ for all $i\neq j$ such that $X_{A}$ is
self dual. Let $k=\sum_{i=1}^{n}k_{i}$ be the total number of repetitions.
Then $A$ is an $s$-pyramidal configuration, with $0\leq s\leq k$. In
particular:
(1) $A$ is a $k$-pyramidal configuration if and only if $k_{i}=0$ for every
${\bf a}_{i}$ free vector of the $C$ configuration and
(2) $A$ is a non-pyramidal configuration if and only if $k_{i}=1$ for every
${\bf a}_{i}$ free vector of the $C$ configuration and $k_{i}=0$ for every
${\bf a}_{i}$ non-free vector of the $C$ configuration.

###### Proof.

By [5, Theorem 3.8] $C$ is $k$-pyramidal, which means it has $k$ free vectors.
We claim that the vector ${\bf a}_{i}$ is free in $A$ if and only if the
vector ${\bf a}_{i}$ is free in $C$ and $k_{i}=0$.
Suppose that the vector ${\bf a}_{i}$ is free in $A$ then there exist ${\bf
u}\in{\mathbb{Z}}^{d}$ such that ${\bf u}\cdot{\bf a}_{i}=1$ for only one
vector of $A$ and ${\bf u}\cdot{\bf a}_{j}=0$ for all the rest. Thus $k_{i}=0$
and ${\bf a}_{i}$ is free in $C$. For the other direction, suppose that the
vector ${\bf a}_{i}$ is free in $C$, then there exist ${\bf
u}\in{\mathbb{Z}}^{d}$ such that ${\bf u}\cdot{\bf a}_{i}=1$ and ${\bf
u}\cdot{\bf a}_{j}=0$ for all $j\not=i$. Then ${\bf u}\cdot{\bf a}_{i}=1$ for
only one vector in $A$, since $k_{i}=0$, and ${\bf u}\cdot{\bf a}_{j}=0$ for
all other vectors in $A$. Thus ${\bf a}_{i}$ is free in $A$.

From the previous claim we conclude that the number of free vectors in $A$ is
less than or equal to the number of free vectors in $C$, thus $s\leq k$. The
equality $s=k$ happens if and only if $k_{i}=0$ for every free vector of $C$.

Suppose now that $A$ is non-pyramidal configuration therefore $k_{i}\geq 1$
for all free vectors ${\bf a}_{i}$. Thus

$k=\sum_{i=1}^{n}k_{i}\geq\sum_{{\bf a}_{i}\text{free}}k_{i}\geq k.$

Therefore $k=\sum_{i=1}^{n}k_{i}=\sum_{{\bf a}_{i}\text{free}}k_{i}=k$ which
means that $k_{i}=0$ if ${\bf a}_{i}$ is a non-free vector and $k_{i}=1$ if
${\bf a}_{i}$ is a free vector of the $C$ configuration. For the converse,
suppose that $k_{i}=1$ if ${\bf a}_{i}$ is a free vector of $C$ then there is
no free vector in $A$ and thus the $A$ configuration is non-pyramidal. ∎

##  4\. Toric bases and combinatorial properties of self-dual toric varieties

A Markov basis of an $m\times n$ matrix $A$ is a finite subset $M$ of
$\operatorname{Ker}_{\mathbb{Z}}(A)$ such that whenever
$\textbf{w},\textbf{v}\in\mathbb{N}^{n}$ and
$\textbf{w}-\textbf{v}\in\operatorname{Ker}_{\mathbb{Z}}(A)$ there exists a
set $\\{\textbf{u}_{i}|i=1,\ldots,l\\}$ that connects w to v, where either
$\textbf{u}_{i}\in M$ or the opposite $-\textbf{u}_{i}\in M$. To connect w to
v means that $\textbf{w}-\textbf{v}=\sum_{i=1}^{l}\textbf{u}_{i}$, and
$(\textbf{w}-\sum_{i=1}^{p}\textbf{u}_{i})\in\mathbb{N}^{n}$ for all $1\leq
p\leq l$. A Markov basis of $A$ is minimal if no proper subset of it is a
Markov basis of $A$. A renowned result of Diaconis and Sturmfels [14, Theorem
3.1] says that $\mathcal{M}$ is a minimal Markov basis of $A$ if and only if
the set $\\{\textbf{x}^{u^{+}}-\textbf{x}^{u^{-}}|\textbf{u}\in\mathcal{M}\\}$
forms a minimal system of binomial generators for the toric ideal $I_{A}$. The
universal Markov basis $\mathcal{M}(A)$ of $A$ is the union of all minimal
Markov bases of $A$, up to identification of opposite vectors. The set of
indispensable elements $\mathcal{S}(A)$ is the intersection of all different
minimal Markov bases, via the same identification. The set of indispensable
binomials $\mathcal{S}(I_{A})$ of the toric ideal is the set
$\\{\textbf{x}^{u^{+}}-\textbf{x}^{u^{-}}|\textbf{u}\in\mathcal{S}(A)\\}$,
that is the intersection of all minimal systems of binomial generators of
$I_{A}$, up to identification of opposite binomials.

We say that ${\bf{u}}={\bf{v}}+_{sc}{\bf{w}}$ is a semiconformal decomposition
of the vector u in two vectors v, w, if u=v+w and $v_{i}>0$ implies $w_{i}\geq
0$ and $w_{i}<0$ implies that $v_{i}\leq 0$ for every $1\leq i\leq n$, a
concept introduced first in [20], where by $v_{i},w_{i}$ we denote the $i$-th
component of the vectors ${\bf v}$ and ${\bf w}$, respectively. We call the
decomposition proper if both ${\bf v},{\bf w}$ are nonzero. From [8,
Proposition 1.1], we know that the set of the indispensable elements
$\mathcal{S}(A)$ of $A$ consists of all nonzero vectors in
$\operatorname{Ker}_{{\mathbb{Z}}}(A)$ with no proper semiconformal
decomposition.

We say that ${\bf u}={\bf v}+_{ssc}{\bf w}$ is a strongly semiconformal
decomposition of the vector ${\mathbb{u}}$ in two vectors v, w, if ${\bf
u}={\bf v}+{\bf w}$, ${\bf u}^{+}>{\bf v}^{+}$ and ${\bf u}^{-}>{\bf w}^{-}$.
Recall that for two vectors ${\bf{u}},{\bf{v}}\in\mathbb{Z}^{n}$ we say that
${\bf{u}}\geq{\bf{v}}$ if ${\bf{u}}-{\bf{v}}\in\mathbb{N}^{n}$, where the
inequality is strict if ${\bf{u}}\neq{\bf{v}}$. Note that strongly
semiconformal decomposition can be of more than two vectors, see [8, Section
1]. The universal Markov basis of $A$, $\mathcal{M}(A)$, consists of all
nonzero vectors in $\operatorname{Ker}_{{\mathbb{Z}}}(A)$ with no proper
strongly semiconformal decomposition, [8, Proposition 1.4].

######  Definition 4.1.

A toric ideal $I_{A}$ is called strongly robust if the Graver basis of $I_{A}$
is equal to a minimal Markov basis of $I_{A}$.

It follows from the definition, as we have seen in the Introduction, that for
strongly robust ideals the following sets are identical: the set of
indispensable elements, any minimal Markov basis, the universal Markov basis,
any reduced Gröbner basis, the universal Gröbner basis and the Graver basis of
$A$.

######  Theorem 4.2.

Let $A=\\{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{n}\\}$ be a non-pyramidal
configuration such that $X_{A}$ is a self-dual projective toric variety. Then
the toric ideal is strongly robust.

###### Proof.

Since $A$ is non-pyramidal configuration then by Theorem 3.4 we obtain
$\sum_{i}(c_{B})_{i}=0$ for every bouquet $B$ of $A$. The latter condition
implies that each vector ${\mathbb{c}}_{B}$ has at least one positive and one
negative component, since by definition ${\mathbb{c}}_{B}$ has the first
nonzero component positive. Thus each bouquet $B$ is mixed by Theorem 3.4, and
applying [24, Corollary 4.4] we obtain that the toric ideal is strongly
robust. ∎

######  Example 4.3.

Let $A=(4\ 5\ 6\ 7)$ and $C$ be the generalized Lawrence matrix defined in the
Example 3.6. Then the matrix $A=(4\ 5\ 6\ 7)$ has 29 elements in the Graver
basis, those are:

$\operatorname{Gr}(A)=\\{(5,-4,0,0),(1,-2,1,0),(2,-3,0,1),(4,-2,-1,0),(3,-1,0,-1),$
$(1,-1,-1,1),(2,1,-1,-1),(1,2,0,-2),(3,0,-2,0),(2,0,1,-2),(5,0,-1,-2),$
$(4,1,0,-3),(7,0,0,-4),(0,1,-2,1),(2,2,-3,0),(1,0,-3,2),(1,3,-2,-1),$
$(1,1,2,-3),(0,4,-1,-2),(0,3,1,-3),(1,4,-4,0),(1,0,4,-4),(1,-5,0,3),$
$(0,5,-3,-1),(0,2,3,-4),(0,6,-5,0),(0,1,5,-5),(0,7,0,-5),(0,0,7,-6)\\}.$

Theorem 4.2 says that $I_{C}$ is strongly robust. The Graver basis by [24,
Theorem 3.7] it has 29 elements all in the form $D({\bf
u})=u_{1}\gamma_{1}+u_{2}\gamma_{2}+u_{3}\gamma_{3}+u_{4}\gamma_{4}=(u_{1},-u_{1},7u_{2},-4u_{2},-3u_{2},2023u_{3},u_{3},-2024u_{3},u_{4},3u_{4},-2u_{4},-2u_{4})$,
where ${\bf u}=(u_{1},u_{2},u_{3},u_{4})\in\operatorname{Gr}(A)$ and
$\gamma_{1}=(1,-1,0,0,0,0,0,0,0,0,0,0),\\\
\gamma_{2}=(0,0,7,-4,-3,0,0,0,0,0,0,0),\
\gamma_{3}=(0,0,0,0,0,2023,1,-2024,0,0,0,0),\\\
\gamma_{4}=(0,0,0,0,0,0,0,0,1,3,-2,-2)$ are the vectors with support
$c^{\prime}_{1},c^{\prime}_{2},c^{\prime}_{3},c^{\prime}_{4}$ of the Example
3.6. For example, for the element $(1,3,-2,-1)\in\operatorname{Gr}(A)$ we have
the corresponding Graver basis element

$D(1,3,-2,-1)=(1,-1,21,-12,-9,-4046,-2,4048,-1,-3,2,2)\in\operatorname{Gr}(C).$

This element, $D(1,3,-2,-1)$, is also indispensable thus belongs to every
Markov basis and every reduced Gröbner basis, as also the other 28 elements in
the form $D({\bf u})$. This property of $I_{C}$ is not true for $I_{A}$, for
example $(1,3,-2,-1)\in\operatorname{Gr}(A)$ is not indispensable, since it
has a proper semi-conformal decomposition:

$(1,3,-2,-1)=(1,2,0,-2)+_{sc}(0,1,-2,1).$

Note that if we drop the assumption of non-pyramidal configuration then the
conclusion is no longer true. However, we still have an important
combinatorial information. The toric ideal is weakly robust. We recall that
the universal Markov basis of $I_{A}$, $\mathcal{M}(A)$, is the union of all
minimal generating sets of $I_{A}$ modulo the $\pm$ sign (see [20, Definition
3.1]) and it has a useful algebraic characterization described in [8,
Proposition 1.4], similar to the conformality for Graver basis.

######  Definition 4.4.

A toric ideal $I_{A}$ is called weakly robust if the Graver basis of $I_{A}$
is equal to the universal Markov basis of $I_{A}$.

We examine the relation between the Graver basis of a multiset configuration
$A$ with ground set $C$ and the Graver basis of $C$. Let
$C=\\{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{n}\\}\subset{\mathbb{Z}}^{d}$ be
such that ${\mathbb{a}}_{i}\neq{\mathbb{a}}_{j}$ for all $i\neq j$ and

$A=\\{\underbrace{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{1}}_{k_{1}+1},\ldots,\underbrace{{\mathbb{a}}_{n},\ldots,{\mathbb{a}}_{n}}_{k_{n}+1}\\}$

a multiset configuration with $k_{1},\ldots,k_{n}$ nonnegative integers.
Denote by $k$ the nonnegative integer $\sum_{i=1}^{n}k_{i}$. If $k=0$ then
$A=C$, thus $\operatorname{Gr}(A)=\operatorname{Gr}(C)$. In the case that
$k\geq 1$, consider $I_{C}\subset K[x_{1},\ldots,x_{n}]$ and

$I_{A}\subset
K[y_{11},\ldots,y_{1k_{1}+1},\ldots,y_{n1},\ldots,y_{nk_{n}+1}].$

Let $\pi$ be the restriction from $\operatorname{Ker}_{{\mathbb{Z}}}(A)$ to
$\operatorname{Ker}_{{\mathbb{Z}}}(C)$ of the linear map

$\pi(u_{11},\ldots,u_{1,k_{1}+1},\ldots,u_{n1},\ldots,u_{n,k_{n}+1})=(\sum_{i=1}^{k_{1}+1}u_{1i},\ldots,\sum_{i=1}^{k_{n}+1}u_{ni})$

from ${\mathbb{Q}}^{n+k}$ to ${\mathbb{Q}}^{n}$.

Note that the vectors in $A$ are in n-blocks. All the vectors in the i-th
block are equal to each other and different from the elements in the other
blocks. The size of the $i-$th block is $k_{i}+1$. Also the components of
${\bf u^{\prime}}$ in $\operatorname{Ker}_{{\mathbb{Z}}}(A)$ are partitioned
in n-blocks. The components in the i-th block are denoted by
$u_{i1},\cdots,u_{ik_{i}+1}$.

For ${\bf u}\in\operatorname{Ker}_{{\mathbb{Z}}}(C)$ we denote by
$\pi^{-1}({\bf u})_{+}$ all elements ${\bf u^{\prime}}$ in
$\operatorname{Ker}_{{\mathbb{Z}}}(A)$ such that $\pi({\bf u^{\prime}})={\bf
u}$ and all nonzero components in the i-block of ${\bf u^{\prime}}$ have the
same sign, for every $i$-block, this means that if $u_{i}>0$ then
$u^{\prime}_{ij}\geq 0$, if $u_{i}=0$ then $u^{\prime}_{ij}=0$ and if
$u_{i}<0$ then $u^{\prime}_{ij}\leq 0$, for all possible $j$. Let $Q_{A}$ be
the set of all the elements ${\bf c}_{is,it}$ in
$\operatorname{Ker}_{{\mathbb{Z}}}(A)$ in the form: all components of ${\bf
c}_{is,it}$ are zero except an 1 in the $is$ position and a $-1$ in the $it$
position, $1\leq s,t\leq k_{i}+1.$

######  Theorem 4.5.

The Graver basis of $I_{A}$ is

$(\bigcup_{{\bf u}\in Gr(C)}\pi^{-1}({\bf u})_{+})\bigcup Q_{A}.$

###### Proof.

The elements in $Q_{A}$ are circuits, since they are homogeneous of degree one
and have support of cardinality two. Therefore they are in the Graver basis,
see [26, Proposition 4.11].

Let ${\bf u}^{\prime}\in\pi^{-1}({\bf u})_{+}$ for some ${\bf u}\in Gr(C)$.
Suppose that ${\bf u}^{\prime}={\bf v}^{\prime}+_{c}{\bf w}^{\prime}$, $\pi$
is linear therefore $\pi({\bf u}^{\prime})=\pi({\bf v}^{\prime})+\pi({\bf
w}^{\prime})$. Note that all non-zero components in the i-block of ${\bf u}$
have the same sign, since ${\bf u}^{\prime}\in\pi^{-1}({\bf u})_{+}$. Then the
conformality of the sum ${\bf u}^{\prime}={\bf v}^{\prime}+_{c}{\bf
w}^{\prime}$ implies that all non-zero components in the i-th blocks of ${\bf
u}^{\prime},{\bf v}^{\prime},{\bf w}^{\prime}$ have the same sign. This imply
that the sum $\pi({\bf u}^{\prime})=\pi({\bf v}^{\prime})+_{c}\pi({\bf
w}^{\prime})$ is conformal. But ${\bf u}=\pi({\bf u}^{\prime})$ is in the
Graver of $C$ therefore either $\pi({\bf v}^{\prime})$ or $\pi({\bf
w}^{\prime})$ is zero. But all the non-zero components in the i-th block have
the same sign, for every $i\in[n]$, therefore if $\pi({\bf v}^{\prime})={\bf
0}$ then ${\bf v}^{\prime}={\bf 0}$ and if $\pi({\bf w}^{\prime})={\bf 0}$
then ${\bf w}^{\prime}={\bf 0}$. Therefore the sum ${\bf u}^{\prime}={\bf
v}^{\prime}+_{c}{\bf w}^{\prime}$ is not proper and ${\bf u}^{\prime}\in
Gr(A)$. Thus $(\bigcup_{{\bf u}\in Gr(C)}\pi^{-1}({\bf u})_{+})\bigcup
Q_{A}\subset Gr(A)$.

Let ${\bf u}^{\prime}\in Gr(A)$ which is not in $Q_{A}$. First we claim that
all non-zero components in any i-th block of ${\bf u}^{\prime}$ are having the
same sign. Suppose not, then let $u^{\prime}_{is}>0$ and $u^{\prime}_{it}<0$
for some $s,t$, then define ${\bf u}^{\prime\prime}$ to have all components
the same as ${\bf u}^{\prime}$ except $is$ and $it$ and
$u^{\prime\prime}_{is}=u^{\prime}_{is}-1\geq 0$,
$u^{\prime\prime}_{it}=u^{\prime}_{it}+1\leq 0$. Then the sum ${\bf
u}^{\prime}={\bf u}^{\prime\prime}+{\bf c}_{is,it}$ is conformal and both
${\bf u}^{\prime\prime},{\bf c}_{is,it}$ are in the kernel of A. Note that
${\bf c}_{is,it}$ has all components zero, except the $is$ which is 1 and the
$it$ which is -1. The ${\bf u}^{\prime}$ is not in $Q_{A}$ thus both ${\bf
u}^{\prime\prime},{\bf c}_{is,it}$ are different from zero. A contradiction,
since ${\bf u}^{\prime}\in Gr(A)$. Therefore all non-zero elements in the i-th
block of ${\bf u}^{\prime}$ are having the same sign, for every $i\in[n]$.

Next we claim that $\pi({\bf u}^{\prime})\in Gr(C)$. Suppose that $\pi({\bf
u}^{\prime})={\bf u}={\bf v}+_{c}{\bf w}$, where ${\bf v},{\bf w}$ are in the
kernel of $C$. Then look at the i-th block of ${\bf u}^{\prime}$ we have $\sum
u^{\prime}_{ij}=u_{i}=v_{i}+w_{i}$ and since the sum is conformal all nonzero
components are of the same sign.

Suppose first that $u_{i}>0$. In the case
$v_{i}=u_{i}=\sum_{j=1}^{k_{i}+1}u^{\prime}_{ij}$ we set
$v_{ij}=u^{\prime}_{ij}$ and $w_{ij}=0$. Otherwise there exist a $t$ such that
$\sum_{j=1}^{t}u^{\prime}_{ij}\leq v_{i}<\sum_{j=1}^{t+1}u^{\prime}_{ij}$ for
$0\leq t\leq k_{i}+1$, where for $t=0$ we set
$\sum_{j=1}^{t}u^{\prime}_{ij}=0$. Set $v_{ij}=u^{\prime}_{ij}$ and $w_{ij}=0$
for $j\leq t$, $v_{it+1}=v_{i}-\sum_{j=1}^{t}u^{\prime}_{ij}\geq 0$ and
$w_{ij}=\sum_{j=1}^{t+1}u^{\prime}_{ij}-v_{i}>0$ and $v_{ij}=0$ and
$w_{ij}=u^{\prime}_{ij}$ for $j>t$.

In the case that $u_{i}=0$ then $v_{i}=0=w_{i}$ from the conformality. We set
all $v_{ij}=0=w_{ij}.$

In the last case $u_{i}<0$. In the case
$v_{i}=u_{i}=\sum_{j=1}^{k_{i}+1}u^{\prime}_{ij}$ we set
$v_{ij}=u^{\prime}_{ij}$ and $w_{ij}=0$. Otherwise there exist a $t$ such that
$\sum_{j=1}^{t}u^{\prime}_{ij}\geq v_{i}>\sum_{j=1}^{t+1}u^{\prime}_{ij}$ for
$0\leq t\leq k_{i}+1$, where for $t=0$ we set
$\sum_{j=1}^{t}u^{\prime}_{ij}=0$. Set $v_{ij}=u^{\prime}_{ij}$ and $w_{ij}=0$
for $j\leq t$, $v_{it+1}=v_{i}-\sum_{j=1}^{t}u^{\prime}_{ij}\leq 0$ and
$w_{ij}=\sum_{j=1}^{t+1}u^{\prime}_{ij}-v_{i}<0$ and $v_{ij}=0$ and
$w_{ij}=u^{\prime}_{ij}$ for $j>t$.

Let ${\bf v}^{\prime}$ the vector with components $v_{ij}$ and ${\bf
w}^{\prime}$ the vector with components $w_{ij}$. Then ${\bf u}^{\prime}={\bf
v}^{\prime}+_{c}{\bf w}^{\prime}$ which implies that either ${\bf
v}^{\prime}={\bf 0}$ or ${\bf w}^{\prime}={\bf 0}$. But from the definition
${\bf v}^{\prime}$ is different from zero, since ${\bf u}^{\prime}\not={\bf
0}$. Thus ${\bf w}^{\prime}={\bf 0}$. But this can happen only if ${\bf
w}={\bf 0}$, which means that the conformal sum $\pi({\bf u}^{\prime})={\bf
v}+_{c}{\bf w}$ is never proper. Therefore $\pi({\bf u}^{\prime})\in Gr(C)$.
Set $\pi({\bf u}^{\prime})={\bf u}$ thus ${\bf u}^{\prime}\in\pi^{-1}({\bf
u})_{+}$ for a ${\bf u}\in Gr(C)$. ∎

######  Theorem 4.6.

Let $A$ be any configuration such that $X_{A}$ is a self-dual projective toric
variety. Then the toric ideal $I_{A}$ is weakly robust.

###### Proof.

If $A$ is non-pyramidal then we are done by theorem 4.2, since strongly robust
implies weakly robust.

Otherwise, by applying [5, Theorem 3.8] since $X_{A}$ is self-dual, we obtain
that the ground set of $A$,
$C=\\{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{n}\\}$, is $k$-pyramidal and,
after a permutation, the vectors in
$C_{f}=\\{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{k}\\}$ are free and the
configuration $C_{nf}=\\{{\mathbb{a}}_{k+1},\ldots,{\mathbb{a}}_{n}\\}$ is
non-pyramidal with the corresponding variety $X_{C_{nf}}$ being self-dual.

Since $(1,1,\cdots,1,1)$ belongs in the row span of the matrix $A$ then
$\operatorname{Ker}_{{\mathbb{Z}}}(A)\cap{\mathbb{N}}^{n}=\\{{\mathbf{0}}\\}$
which implies that $\mathcal{M}(A)\subset Gr(A)$, see [9, Theorem 2.3].

Next we prove that $\operatorname{Gr}(A)\subset\mathcal{M}(A)$. Note that an
element ${\bf u}^{\prime}$ belongs in the universal Markov basis,
$\mathcal{M}(A)$, if and only if the corresponding binomial $y^{{\bf
u}^{\prime}_{+}}-y^{{\bf u}^{\prime}_{-}}$ cannot be written as a combination
of binomials of $I_{A}$ of strictly smaller $A$-degree, see [7, Section 2] or
[15, Theorem 1.3.2] or [8, Proposition 1.4]. The $A$-degree of the binomial
$y^{{\bf u}^{\prime}_{+}}-y^{{\bf u}^{\prime}_{-}}$ is $A{\bf
u}^{\prime}_{+}(=A{\bf u}^{\prime}_{-}).$ Let ${\bf
u}^{\prime}\in\operatorname{Gr}(A)$ such that ${\bf
u}^{\prime}\notin\mathcal{M}(A)$. Then we have

$y^{{\bf u}^{\prime}_{+}}-y^{{\bf u}^{\prime}_{-}}=\sum y^{{\bf
m}_{i}}(y^{{{\bf u}_{i}}_{+}}-y^{{{\bf u}_{i}}_{-}})\in
K[y_{11},\ldots,y_{1k_{1}+1},\ldots,y_{n1},\ldots,y_{nk_{n}+1}],$

where $y^{{\bf m}_{i}}\not=1$, for some ${\bf
u}_{i}\in\operatorname{Ker}_{{\mathbb{Z}}}(A)$. Certainly no element in
$Q_{A}$ can be written in this form, since all are of degree 1. Therefore
${\bf u}^{\prime}\in\pi^{-1}({\bf u})_{+}$ for some ${\bf u}\in Gr(C)$, by
Theorem 4.5. Note that the vectors ${\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{k}$
are free therefore the first $k$ components are zero for every element ${\bf
v}$ in $\operatorname{Ker}_{{\mathbb{Z}}}(C)$, thus $supp({\bf
v})\subset\\{k+1,\cdots,n\\}$. Let $p$ be the projection from
${\mathbb{Z}}^{n}$ to ${\mathbb{Z}}^{n-k}$ to the last $n-k$ components. Then
$p({\bf v})\in\operatorname{Ker}_{{\mathbb{Z}}}(C_{nf})$ for every ${\bf
v}\in\operatorname{Ker}_{{\mathbb{Z}}}(C)$ and for the ${\bf u}\in Gr(C)$ we
have $p({\bf u})\in Gr(C_{nf})$, since conformal decompositions in
$\operatorname{Ker}_{{\mathbb{Z}}}(C_{nf})$ give conformal decompositions in
$\operatorname{Ker}_{{\mathbb{Z}}}(C)$.

Then setting in the above equation $y_{ij}=x_{i}$ for $k+1\leq i\leq n$,
$1\leq j\leq k_{i}+1$, and $y_{ij}=0$ for $1\leq i\leq k$, $1\leq j\leq
k_{i}+1$, we have

$x^{{\bf u}_{+}}-x^{{\bf u}_{-}}=x^{p({\bf u})_{+}}-x^{p({\bf u})_{-}}=\sum
x^{\pi({\bf m}_{i})}(x^{p({\pi({{\bf u}_{i}}_{+})})}-x^{p({\pi({{\bf
u}_{i}}_{-})})})$

in the ring $K[x_{k+1},\ldots,x_{n}]$, with all $x^{\pi({\bf m}_{i})}\not=1$.
Therefore $p({\bf u})\notin\mathcal{M}(C_{nf})$. Note that the sum with the
$x$’s in the right part of the equation above may have less terms than the
previous one with the $y$’s, but still has at least one, since the left part
is not zero.

The $X_{C_{nf}}$ is a non-pyramidal self-dual variety thus from Theorem 4.2
the ideal $I_{C_{nf}}$ is strongly robust, and so also weakly robust, thus
$\operatorname{Gr}(C_{nf})=\mathcal{M}(C_{nf})$. But $p({\bf u})\in
Gr(C_{nf})$ and $p({\bf u})\notin\mathcal{M}(C_{nf})$, a contradiction.

We conclude that $\operatorname{Gr}(A)=\mathcal{M}(A)$. ∎

Next Theorem proves that for self-dual toric varieties the strongly robust
property is equivalent with the property of being non-pyramidal.

######  Theorem 4.7.

Let $A$ be any configuration such that $X_{A}$ is a self-dual projective toric
variety. The toric ideal $I_{A}$ is strongly robust if and only if $A$ is non-
pyramidal.

###### Proof.

By Theorem 4.2 if $A$ is non-pyramidal then $I_{A}$ is strongly robust.
Suppose that $A$ is $s$-pyramidal with $s\geq 1$. Then the gound set $C$ is
$k$-pyramidal $k\geq s$, by Theorem 3.11.

Suppose that there exist $i$ such that $k_{i}>1$, that means the vector ${\bf
a}_{i}$ appears at least three times in $A$. Then the three circuits ${\bf
c}_{i1,i2}$, ${\bf c}_{i1,i3}$, ${\bf c}_{i2,i3}$ by Theorem 4.5 belong to the
Graver basis of $A$. Note that ${\bf c}_{i1,i3}={\bf c}_{i1,i2}+_{sc}{\bf
c}_{i2,i3}$ is a proper semiconformal decomposition of ${\bf c}_{i1,i3}$, see
introduction of Section 4 or [8, Proposition 1.1], thus ${\bf c}_{i1,i3}$ is
not indispensable. But ${\bf c}_{i1,i3}$ is a circuit and therefore belongs to
the Graver basis of $I_{A}$. For strongly robust ideals every Graver basis
element is indispensable. Thus $I_{A}$ is not strongly robust.

Therefore all $k_{i}$ take the values one or zero. Since $A$ is a pyramidal
configuration by Theorem 3.11 there exist an $i$ such that ${\bf a}_{i}$ is
non free in the $C$ configuration and $k_{i}=1$. Since ${\bf a}_{i}$ is non
free vector in $C$ there exist a ${\bf u}\in Gr(C)$ such that the $i$-th
component of ${\bf u}$ is different from zero, and thus w.l.o.g. we can assume
that it is positive. Consider ${\bf u}^{\prime}\in\pi^{-1}({\bf u})_{+}$ with
$u^{\prime}_{i1}=u_{i}$ and thus $u^{\prime}_{i2}=0$, and ${\bf
v}^{\prime}\in\pi^{-1}({\bf u})_{+}$ with $v^{\prime}_{i1}=u_{i}-1$ and thus
$v^{\prime}_{i2}=1$ then ${\bf u}^{\prime}={\bf c}_{i1,i2}+_{sc}{\bf
v}^{\prime}$ is a semiconformal decomposition of ${\bf u}^{\prime}$ and it is
proper since ${\bf c}_{i1,i2}\not={\bf 0}$ and ${\bf
u}^{\prime}\in\pi^{-1}({\bf u})_{+}$ which implies that ${\bf
u}^{\prime}\not={\bf c}_{i1,i2}$ and ${\bf v}^{\prime}\not={\bf 0}$. Thus
${\bf u}^{\prime}$ is not indispensable, see [8, Proposition 1.1], but it is
in the Graver of $A$, see Theorem 4.5. Therefore $I_{A}$ is not strongly
robust. ∎

######  Example 4.8.

Let $C_{1}$ be the matrix of the Example 3.10. Let $E_{1}$ be the matrix in
which we repeat once all first four columns of $C_{1}$ then Theorem 3.11 says
that the columns of $E_{1}$ form a non-pyramidal configuration. From Theorem
4.5 we have that there are 29+4 elements in the Markov basis which is also
universal Gröbner and Graver, as was expected from Theorem 4.2 since it is
non-pyramidal self dual. The 29 elements have the first eight components zero
and the rest are the components of the 29 elements of $I_{C}$, in the form
$D(u)$ described above shifted by eight places. The other four elements are
the vectors $c_{11,12},c_{21,22},c_{31,32},c_{41,42}$ corresponding to the
binomials $y_{11}-y_{12},y_{21}-y_{22},y_{31}-y_{32},y_{41}-y_{42}$.

######  Example 4.9.

In this example let $E_{2}$ be a 16+4 multiset configuration with the ground
set the set of columns of $C_{1}$ where we repeat four times only the column
12 of $C_{1}$ (five columns in total identical with the 12-th column) then we
get a 4-pyramidal self dual toric variety according to Theorem 3.11. Theorem
4.6 says that the universal Markov basis and the Graver basis coincide and
their elements are described in Theorem 4.5. They consist of 10 elements in
the form $c_{ij,ik}$, where $i=12$ and $j,k$ take different values from 1 to
5. The rest of the elements are the elements in the sets $\pi^{-1}({\bf
u})_{+}$ where ${\bf u}$ takes the 29 values from the Graver basis of
$I_{C_{1}}$. Then if the absolute value of the 12-th component of a given
vector ${\bf u}$ is $m$ there are

$f(m)=\left(\begin{array}[]{c}m+4\\\ 4\end{array}\right)$

elements in the set $\pi^{-1}({\bf u})_{+}$, since this is the number of
monomials in 5 variables of degree $m$. There are 8 elements in the Graver
basis of $I_{C_{1}}$ with the $12$-th component $0$, there are $8$
$(4,4,2,2,1)$ elements with the absolute value of the $12$-th component $2024$
$(4048,6072,8096,10120,14168$ correspondingly). Therefore there are

$\omega=10+8f(0)+8f(2024)+4f(4048)+4f(6072)+2f(8096)+2f(10120)+f(14168)$

elements in the Graver basis $I_{E_{2}}$, which is also universal Markov by
Theorem 4.6. From those we need only 33 (=29+4) elements in any minimal Markov
basis, four from the 10 in the form $c_{ij,ik}$ and one from each fiber
$\pi^{-1}({\bf u})_{+}$, ${\bf u}\in Gr(C_{1})$.

Let $G$ be a reduced Gröbner basis of $I_{E_{2}}$ with respect to a monomial
order $>$. Let $y_{12,k}$ be the smallest variable among
$y_{12,1},y_{12,2},y_{12,3},y_{12,4},y_{12,5}$, in the order $>$. Then all
four elements $y_{12,i}-y_{12,k}$, where $i\not=k$, are in $G$. Therefore from
each fiber $\pi^{-1}({\bf u})_{+}$, such that ${\bf u}$ has $u_{12}$ component
nonzero the only element that belongs to $G$ is the one that the 12th block is
only $y_{12,k}^{u_{12}}$. Therefore from each fiber like that $\pi^{-1}({\bf
u})_{+}$ there are exactly five binomials that can appear in the universal
Gröbner basis of $I_{E_{2}}$, in each of these five only one variable from
$y_{12,i}$ appears. There are 21 elements in the Graver basis of $I_{C_{1}}$
with the 12th $u_{12}$ component nonzero therefore the cardinality of the
universal Gröbner basis is $10+8+5\cdot 21=123$.

There is huge difference between the size of a Markov basis and the universal
Markov basis (33 to $\omega$) but also between the size of the universal
Gröbner basis and the Graver basis (123 to $\omega$). In $\omega$ only the
$f(14168)$ is about $1.68\cdot 10^{15}.$

Strongly robust ideals have a unique Markov basis while the weakly robust
ideals may have a huge number of different minimal Markov bases. Let $\Omega$
denote the number of different minimal Markov bases of $I_{E_{2}}$. Applying
to $I_{E_{2}}$ the formula of [7, Theorem 2.9] for the number of different
minimal Markov bases of a toric ideal $I_{A}$, we obtain

$\Omega=5^{3}f(0)^{8}f(2024)^{8}f(4048)^{4}f(6072)^{4}f(8096)^{2}f(10120)^{2}f(14168).$

Note that $5^{3}$ is the number of spanning trees on the complete graph on 5
vertices for the Betti fiber that has vertices
$y_{12,1},y_{12,2},y_{12,3},y_{12,4},y_{12,5}$. The other Betti fibers have
$f(m)+1$ elements, for appropriate $m$, in two connected components, one with
$f(m)$ elements and the other with just one element, see [7, Section 2] or
[15, Theorem 1.3.2] for details.

Acknowledgements. The second author has been partially supported by the grant
PN-III-P4-ID-PCE-2020-0029, within PNCDI III, financed by Romanian Ministry of
Research and Innovation, CNCS - UEFISCDI.

## References

  * [1] Bernstein, D., Zelevinsky, A., Combinatorics of maximal minors, J. Algebr. Comb. 2(2), 111–121 (1993).
  * [2] Boocher, A., Free resolutions and sparse determinantal ideals, Math. Res. Letters 19(4) 805–821 (2012).
  * [3] Boocher, A., Robeva, E., Robust toric ideals, J. Symbolic Comput., 68(1), 254–264 (2015).
  * [4] Boocher, A., Brown, B. C., Duff, T., Lyman, L. , Murayama, T., Nesky, A., and Schaefer, K., Robust graph ideals, Ann. Comb. 19(4), 641–660 (2015).
  * [5] M. Bourel, A. Dickenstein, A. Rittatore, Self-dual projective toric varieties, J. London Math. Soc. (2) 84 (2011), 514–540.
  * [6] E. Cattani, A. Dickenstein, Non-splitting Flags, iterated circuits, $g$-Matrices and Cayley configurations, Vietnam Journal of Mathematics 50 (3) 679–706 (2022).
  * [7] H. Charalambous, A. Katsabekis, A. Thoma, Minimal systems of binomial generators and the indispensable complex of a toric ideal, Proc. Amer. Math. Soc. 135, 3443–3451 (2007).
  * [8] H. Charalambous, A. Thoma, M. Vladoiu, Markov complexity of monomial curves, J. Algebra 417 391–411 (2014).
  * [9] H. Charalambous, A. Thoma, M. Vladoiu, Markov bases and generalized Lawrence liftings, Annals of Combinatorics 19(4), 661–669 (2015).
  * [10] Conca, A., De Negri, E., Gorla, E., Universal Gröbner bases for maximal minors, International Mathematics Research Notices 11, 3245–3262 (2015).
  * [11] Conca, A., De Negri, E., Gorla, E., Cartwright-Sturmfels ideals associated to graphs and linear spaces. J. Comb. Algebra 2 (3), 231–257 (2018).
  * [12] Curran, R., Cattani, E., Restriction of $A$-Discriminants and dual defect toric varieties, J. Symbolic Comput. 42 (1-2) 115–135 (2007).
  * [13] Dickenstein, A., Feichtner, E.M., Sturmfels, B., Tropical Discriminants, J. Amer. Math. Soc. 20 (4), 1111–1133 (2007).
  * [14] P. Diaconis, B. Sturmfels, Algebraic algorithms for sampling from conditional distributions, Annals of Statistics 26(1), 363–397 (1998).
  * [15] M. Drton, B. Sturmfels, S. Sullivant, Lectures on algebraic statistics, Oberwolfach Seminars, 39. Birkhäuser Verlag, Basel, viii+171 pp (2009).
  * [16] Esterov, A., Characteristic classes of affine varieties and Plücker formulas for affine morphisms, J. Eur. Math. Soc. 20 (1) 15–59 (2018).
  * [17] K. Furukawa, A. Ito, Combinatorial description of Dual Defects of Toric Varieties, Commun. Contemp. Math. 23(1) 2050001, 31pp. (2021).
  * [18] Garcia-Marco, I., Tatakis, Ch., On robustness and related properties on Toric ideals, J. Algebr. Comb. 57, 21–52 (2023).
  * [19] Gross, E., Petrović, S., Combinatorial degree bound for toric ideals of hypergraphs, Internat. J. Algebra Comput. 23 (6) 1503–1520 (2013).
  * [20] S. Hoşten, S. Sullivant, A finiteness theorem for Markov bases of hierarchical models, J. Combin. Theory Ser. A 114, 311–321 (2007).
  * [21] D. Kosta, A. Thoma, M. Vladoiu, On the strongly robust property of toric ideals, J. Algebra 616, 1–25 (2023).
  * [22] D. Kosta, A. Thoma, M. Vladoiu, The strongly robust simplicial complex of monomial curves, arXiv:2305.11743.
  * [23] S. Petrović, D. Stasi, Toric algebra of hypergraphs, J. Algebr. Comb. 39, 187–-208 (2014).
  * [24] S. Petrović, A. Thoma, M. Vladoiu, Bouquet algebra of toric ideals, Journal of Algebra, 512 493–525 (2018).
  * [25] S. Petrović, A. Thoma, M. Vladoiu, Hypergraph encodings of toric ideals, J. Combin. Theory Ser. A, 166 11–41 (2019).
  * [26] B. Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series, No. 8 American Mathematical Society Providence, R.I. (1995).
  * [27] Sturmfels, B., Zelevinsky, A., Maximal minors and their leading terms, Adv. Math. 98(1), 65–112 (1993).
  * [28] S. Sullivant, Strongly robust toric ideals in codimension 2, Journal of Algebraic Statistics, 10 128–136 (2019).
  * [29] Tatakis, Ch., Generalized robust toric ideals, J. Pure Appl. Algebra 220, 263–277 (2016).