A software package to compute automorphisms of graded algebras

TL;DR

This paper introduces autgradalg.lib, a Singular library for computing automorphisms of graded algebras, with applications to Mori dream spaces and Fano varieties.

Contribution

It provides an implementation of algorithms for automorphism computation of graded algebras, extending their application to Mori dream spaces and Fano varieties.

Findings

Successfully computed automorphism groups of several Fano varieties.

Demonstrated the library's effectiveness on integral, finitely generated graded algebras.

Extended the applicability of automorphism algorithms to new classes of algebraic varieties.

Abstract

We present a library autgradalg.lib for the free computer algebra system Singular to compute automorphisms of integral, finitely generated $\mathbb{C}$-algebras that are graded pointedly by a finitely generated abelian group. It implements the algorithms developed in 'Computing automorphisms of Mori dream spaces'. We apply the algorithms to Mori dream spaces and investigate the automorphism groups of a series of Fano varieties.

Tables

$X_i$	$\mathrm{Aut}\left({\Omega }_S\right)$	$dim\left(G_i\right)$	$\left[G_i:G_i^0\right]$	$dim\left(\mathrm{Aut}\left(X_i\right)\right)$
$X_3$	$\mathbb{Z}/4\mathbb{Z}$	$3$	$4$	$2$
$X_6$	$\left\{1\right\}$	$5$		$4$
$X_7$	$\left\{1\right\}$	$5$		$4$
$X_{10}$	$\left\{1\right\}$	$4$	$1$	$3$
$X_{12}$	$\left\{1\right\}$	$6$		$5$
$X_{13}$	$\left\{1\right\}$	$4$	$1$	$3$
$X_{14}$	$\left\{1\right\}$	$3$		$2$
$X_{15}$	$\left\{1\right\}$	$5$		$4$
$X_{16}$	$\left\{1\right\}$	$3$		$2$
$X_{18}$	$\left\{1\right\}$	$6$		$5$
$X_{19}$	$\left\{1\right\}$	$4$	$1$	$3$
$X_{20}$	$\left\{1\right\}$	$5$		$4$
$X_{21}$	$\left\{1\right\}$	$3$	$2$	$2$
$X_{25}$	$\left\{1\right\}$	$4$	$1$	$3$
$X_{26}$	$\mathbb{Z}/2\mathbb{Z}$	$3$		$2$
$X_{28}$	$\left\{1\right\}$	$4$	$1$	$3$
$X_{33}$	$\left\{1\right\}$	$6$	$2$	$5$
$X_{34}$	$\left\{1\right\}$	$6$	$2$	$5$
$X_{36}$	$\left\{1\right\}$	$5$	$1$	$4$
$X_{37}$	$\left\{1\right\}$	$4$	$2$	$3$
$X_{38}$	$\left\{1\right\}$	$4$	$3$	$3$
$X_{39}$	$\left\{1\right\}$	$3$		$2$
$X_{40}$	$\left\{1\right\}$	$3$	$1$	$2$
$X_{42}$	$\left\{1\right\}$	$3$	$2$	$2$
$X_{45}$	$\left\{1\right\}$	$4$	$2$	$3$
$X_{46}$	$\left\{1\right\}$	$4$	$1$	$3$
$X_{47}$	$\left\{1\right\}$	$3$	$1$	$2$

Equations

$$R=\underset{w\in K}{⨁}{R}_w\text{with}f{f}^{\prime }\in R_{w+w^{\prime }}\text{for all f∈Rw, f′∈Rw′.}$$

$$R=S/I,S:=\mathbb{C}[T_1,\dots ,T_r]I:=⟨g_1,\dots ,g_s⟩\subseteq S.$$

$$R=S/I,S:=\mathbb{C}[T_1,\dots ,T_8],I:=⟨T_1{T}_6+T_2{T}_5+T_3{T}_4+T_7{T}_8⟩,$$

$$\displaystyle Q\ :=\ \left[\mbox{\tiny$\begin{array}[]{rrrrrrrr}1&1&0&0&-1&-1&2&-2\\ 0&1&1&-1&-1&0&1&-1\\ 1&1&1&1&1&1&1&1\\ \overline{1}&\overline{0}&\overline{1}&\overline{0}&\overline{1}&\overline{0}&\overline{1}&\overline{0}\end{array}$}\right].$$

$$\mathrm{GL}(n)\times S\to S,(A,f)\mapsto A\cdot f:=f({\phi }_A(T_1),\dots ,{\phi }_A(T_r)).$$

$$\mathrm{Aut}({\Omega }_S):=\{\psi \in \mathrm{Aut}(K);\psi ({\Omega }_S)={\Omega }_S\}\subseteq \mathrm{Aut}(K).$$

$$\displaystyle{\rm Aut}(\Omega_{S})\ =\ \Bigg{\{}{\rm id},\left[\mbox{\tiny$\begin{array}[]{rrrr}1&-2&0&0\\ 0&-1&0&0\\ 0&0&1&0\\ 0&1&0&1\\ \end{array}$}\right],$$

$$\displaystyle\left[\mbox{\tiny$\begin{array}[]{rrrr}-1&2&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&1&1&1\\ \end{array}$}\right],\left[\mbox{\tiny$\begin{array}[]{rrrr}-1&0&0&0\\ 0&-1&0&0\\ 0&0&1&0\\ 0&0&1&1\\ \end{array}$}\right]\Bigg{\}}.$$

$$\begin{array}[]{rrrrrrrr}Y(1)&0&0&0&0&0&0&0\\ 0&0&0&0&Y(13)&0&0&0\\ 0&0&0&0&0&0&0&Y(24)\\ 0&0&0&0&0&0&Y(31)&0\\ 0&Y(34)&0&0&0&0&0&0\\ 0&0&0&0&0&Y(46)&0&0\\ 0&0&0&Y(52)&0&0&0&0\\ 0&0&Y(59)&0&0&0&0&0\end{array}$$

$$Y(2),Y(3),\dots ,Y(63),Y(64),-Y(1)Y(13)Y(24)Y(31)Y(34)Y(46)Y(52)Y(59)Z-1$$

$${\mathrm{Stab}}_I({\mathrm{Aut}}_K(S)):=\{A\in {\mathrm{Aut}}_K(S);A\cdot I=I\}\subseteq {\mathrm{Aut}}_K(S).$$

$${\mathrm{Stab}}_I({\mathrm{Aut}}_K(S))\cong {\mathrm{Aut}}_K(R).$$

$$J^{\prime }:=⟨{\ell }_i\left(Y\cdot h_j\right);1\le i\le m,1\le j\le l⟩\subseteq S^{\prime }.$$

$$Y(2),Y(3),\dots Y(63),Y(64),-Y(1)Y(13)Y(24)Y(31)Y(34)Y(46)Y(52)Y(59)Z-1,$$

$$-Y(24)Y(31)+Y(52)Y(59),Y(13)Y(34)-Y(52)Y(59),-Y(13)Y(34)+Y(1)Y(46)$$

$$\mathrm{Cox}(X)=\underset{[D]\in \mathrm{Cl}(X)}{⨁}\Gamma (X,\mathcal{O}(D)),$$

$$\mathrm{Spec}(R)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Aut}}_{\mathrm{Cl}(X)}(\mathrm{Cox}(X))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1. Introduction and setting

Consider an integral, finitely generated ${\mathbb{C}}$-algebra $R$ that is graded by a finitely generated abelian group $K$, i.e., we have a decomposition

[TABLE]

Let the grading to be effective, i.e., the monoid $\vartheta_{R}\subseteq K$ of all $w\in K$ with $R_{w}\neq\{0\}$ generates $K$ as a group, and pointed: this means that we have $R_{0}={\mathbb{C}}$ and the polyhedral cone in $K\otimes{\mathbb{Q}}$ generated by $\vartheta_{R}$ is pointed.

We are interested in the automorphism group ${\rm Aut}_{K}(R)$: it consists of all pairs $(\varphi,\psi)$ such that $\varphi\colon R\to R$ is an automorphism of ${\mathbb{C}}$-algebras, $\psi\colon K\to K$ is an automorphism of groups and $\varphi(R_{w})=R_{\psi(w)}$ holds for all $w\in K$. Note that ${\rm Aut}_{K}(R)$ not only is an important invariant of the algebra $R$, the methods to compute it can by applied to compute symmetries of homogenous ideals $I$. Once given explicitely, the knowledge of the latter largely accelerates further computations involving $I$, see [11, 5, 14] for examples.

This note presents an implementation autgradalg.lib of the algorithms from [10] to compute ${\rm Aut}_{K}(R)$. It is written for the free computer algebra system Singular [7] and is available at [13]. In Section 2, we describe the algorithm [10] to compute ${\rm Aut}_{K}(R)$ and explain our implementation by a series of examples. In Section 3, we apply our implementation to Mori dream spaces. As a result, we determine in Proposition 3.1 information on the automorphism groups of a class of Fano threefolds listed in [3].

2. Automorphisms of graded algebras

Let us fix the assumptions on the algebra $R$ for our algorithms. Firstly, we assume the grading group $K$ to be of shape ${\mathbb{Z}}^{k}\oplus{\mathbb{Z}}/a_{1}{\mathbb{Z}}\oplus\ldots\oplus{\mathbb{Z}}/a_{l}{\mathbb{Z}}$. In particular, $k$ and the list $a_{1},\ldots,a_{l}\in{\mathbb{Z}}_{>1}$ encode $K$. The $K$-grading is determined by the degree matrix $Q=[q_{1},\ldots,q_{r}]$ which has the $q_{i}:=\deg(T_{i})$ as its columns. Moreover, we expect $R$ to be given explicitly in term of generators and relations:

[TABLE]

As one can remove linear equations, it is no restriction to assume that $R$ is minimally presented, i.e., $I\subseteq\langle T_{1},\ldots,T_{r}\rangle^{2}$ holds and the generating set $\{g_{1},\ldots,g_{s}\}$ for $I$ is minimal. From an implementation point of view, it is convenient to impose the following slight restrictions:

• •

the homogenous components $I_{q_{1}},\ldots,I_{q_{r}}$ are all trivial,

• •

the set $\{q_{1}^{0},\ldots,q_{r}^{0}\}\subseteq{\mathbb{Z}}^{k}$ of the free parts $q_{i}^{0}\in{\mathbb{Z}}^{k}$ of the $q_{i}$ contains a lattice basis for ${\mathbb{Z}}^{k}$.

Example 2.1** (autgradalg.lib I).**

Consider the following $K:={\mathbb{Z}}^{3}\oplus{\mathbb{Z}}/2{\mathbb{Z}}$-graded ${\mathbb{C}}$-algebra $R$ from [9, 12, Example 2.1] where

[TABLE]

Then the $K$-grading given by $Q$ is effective and pointed as hinted in the following picture. To use autgradalg.lib, download it from [13] and start Singular in the same directory. We enter $R$ with the commands

LIB "gfanlib.so"; // for cones 2. >

LIB "new_autgradalg.lib"; 3. >

intmat Q[4][8] = 4. >

1,1,0,0,-1,-1,2,-2, 5. >

0,1,1,-1,-1,0,1,-1, 6. >

1,1,1,1,1,1,1,1, 7. >

1,0,1,0,1,0,1,0; 8. >

list TOR = 2; // torsion part of K 9. >

ring S = 0,T(1..8),dp; 10. >

setBaseMultigrading(Q); // grading

q_{1}$$q_{4}$$q_{5}$$q_{6}$$q_{7}$$q_{8}$$(0,0,0)$$q_{2}$$q_{3}

Let us recall shortly the steps of the algorithm to compute ${\rm Aut}_{K}(R)$; for details, we refer to [10]. The overall idea is to present ${\rm Aut}_{K}(R)$ as a stabilizer in the automorphism group ${\rm Aut}_{K}(S)$ of the $K$-graded polynomial ring $S$. In a first step, we will compute a presentation ${\rm Aut}_{K}(S)\subseteq{\rm GL}(n)$ for some $n\in{\mathbb{Z}}_{\geq 1}$. The set $\Omega_{S}:=\{q_{1},\ldots,q_{r}\}$ of generator weights will play a major role. We make use of the following ${\rm GL}(n)$-action.

Construction 2.2**.**

See [10, Construction 3.3]. Write $\Omega_{S}=\{w_{1},\ldots,w_{s}\}$. Determine a ${\mathbb{C}}$-vector space basis $\mathcal{B}_{i}$ for $S_{w_{i}}$ consisting of monomials. Then the concatenation $\mathcal{B}:=(\mathcal{B}_{1},\ldots,\mathcal{B}_{s})$ is a basis for $V=\bigoplus_{i}S_{w_{i}}$. With $n:=|\mathcal{B}|$, in terms of $\mathcal{B}$, each $A\in{\rm GL}(n)$ defines a linear map $\varphi_{A}\colon V\to V$. We obtain an algebraic action

[TABLE]

For the second step, the idea is to determine equations cutting out those matrices in ${\rm GL}(n)$ that permute the homogeneous components $S_{w}$ of same dimension where $w\in\Omega_{S}$. As $\Omega_{S}$ must be fixed by each automorphism, it suffices to consider the finite set

[TABLE]

It can be computed by tracking a lattice basis among the set of free parts $q_{i}^{0}$ of the $q_{i}$, see [10, Remark 3.1].

Algorithm 2.3** (Compute ${\rm Aut}_{K}(S)$).**

See [10, Algorithm 3.7]. *Input: * the $K$-graded polynomial ring $S$.

• •

Determine $\Omega_{S}=\{w_{1},\ldots,w_{s}\}$. Compute a basis $\mathcal{B}$ as in Construction 2.2.

• •

Define the polynomial ring $S^{\prime}:={\mathbb{C}}[Y_{ij};\,1\leq i,j\leq n]$.

• •

Compute an ideal $J\subseteq S^{\prime}$ whose equations ensure for each $A\in V(J)\subseteq{\rm GL}(n)$ the multiplicative condition $A\cdot(f_{1}f_{2})=(A\cdot f_{1})(A\cdot f_{2})$ where $f_{i}\in S$.

• •

Compute ${\rm Aut}(\Omega_{S})\subseteq{\rm Aut}(K)$. Determine the subset $\Gamma_{0}\subseteq{\rm Aut}(\Omega_{S})$ of those $B$, that map $\mathcal{B}_{i}$ bijectively to $\mathcal{B}_{j}$ where $w_{j}=B\cdot w_{i}$.

• •

For each $B\in\Gamma_{0}$, do

• –

compute an ideal $J_{B}\subseteq S^{\prime}$ ensuring that each matrix in $V(J_{B})\subseteq{\rm GL}(n)$ maps the component $S_{w}$ to the component $S_{B\cdot w}$ where $w\in\Omega_{S}$.

• –

Redefine $J:=J\cdot J_{B}$.

*Output: * the ideal $J\subseteq S^{\prime}$. Then $V(J)\subseteq{\rm GL}(n)$ is an algebraic subgroup isomorphic to ${\rm Aut}_{K}(S)$.

Remark 2.4**.**

1. (i)

Note that the third step of Algorithm 2.3 is a finite one, see [10, Definition 3.4(i)] for the precise definition. 2. (ii)

The ring $S^{\prime}$ in Algorithm 2.3 is $K$-graded by defining $\deg(Y_{ij})$ as the degree of the $i$-the element of $\mathcal{B}$. 3. (iii)

The isomorphism $S\to S$ given by $A=(a_{ij})\in V(J)\subseteq{\rm GL}(n)$ is as in Construction 2.2; explictly, it is given by $T_{i}\mapsto\sum_{j}a_{ij}(\mathcal{B}_{i})_{j}.$

Example 2.5** (autgradalg.lib II).**

Let us apply Algorithm 2.3 to Example 2.1. Here, $\mathcal{B}=(T_{1},\ldots,T_{8})$ and all bases $\mathcal{B}_{i}=(T_{i})$ are one-dimensional. Since no weight appears multiple times, $\Omega_{S}=\{q_{1},\ldots,q_{8}\}$. Next, the algorithm will compute ${\rm Aut}(\Omega_{R})$. In our implemenation one can also trigger this step manually if desired:

list origs = autGenWeights(Q, TOR);

The result, origs, is a list of four integral matrices (intmats) standing for the automorphisms of the generator weights

[TABLE]

q_{1}$$q_{4}$$q_{5}$$q_{6}$$q_{7}$$q_{8}$$(0,0,0)$$q_{2}$$q_{3}

Note that ${\rm Aut}(\Omega_{R})$ is isomorphic to the symmetry group ${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$ of a $2$-dimensional rhombus. We now compute ${\rm Aut}_{K}(S)$ with the command

def Sprime = autKS(TOR); 2. >

setring Sprime;

A closer inspection shows that Sprime stands for the ring $S^{\prime}={\mathbb{Q}}[Y_{1},\ldots,Y_{64},Z]$. Furthermore, a list listAutKS will be exported: each element is a triple $(A_{B},B,J_{B})$ where $B$ runs through the four elements of ${\rm Aut}(\Omega_{R})$ and $A_{B}$ is a formal matrix over Sprime that encodes isomorphisms of $S$ as in Remark 2.4(iii). For instance, for listAutKS[2], the sendond entry in the triple $(A_{B},B,J_{B})$ is the second matrix listed in (5) and the matrix $A_{B}$ is

print(listAutKS[2][1]);

[TABLE]

The equations obtained from the zero-entries in $A_{B}$ and its invertible-condition are stored in the ideal $J_{B}$. The third entry is:

print(listAutKS[2][3]);

[TABLE]

Moreover, an ideal Iexported, called $J$ in in Algorithm 2.3, is being exported that is the product over all the ideals $J_{B}$ where $B$ runs through ${\rm Aut}(\Omega_{R})$. This means ${\rm Aut}_{K}(S)\cong S^{\prime}/J$ is isomorphic to Sprime modulo Iexported; the degree matrix of Sprime can be obtained via getVariableWeights().

We come to ${\rm Aut}_{K}(R)$. Restricting the group action of Construction 2.2 to ${\rm Aut}_{K}(S)\subseteq{\rm GL}(n)$, we have an algebraic subgroup given as stabilizer

[TABLE]

Provided $I_{w}=\{0\}$ holds for all $w\in\Omega_{S}$, in [10] the authors have shown that we have an isomorphism

[TABLE]

The final step then is the following. Define the set $\Omega_{I}:=\{\deg(g_{1}),\ldots,\deg(g_{s})\}$ of ideal generator degrees. The idea is to compute (linear) equations ensureing that the vector spaces $I_{u}$, where $u\in\Omega_{I}$, are mapped to one-another.

Algorithm 2.6** (Computing ${\rm Aut}_{K}(R)$).**

See [10, Algorithm 3.8]. Input: the $K$-graded polynomial ring $S$ and the defining ideal $I\subseteq S$ of $R$.

• •

Let $J\subseteq S^{\prime}:={\mathbb{C}}[Y_{ij};\;1\leq i,j\leq n]$ be the output of Algorithm 2.3.

• •

Compute $\Omega_{I}$ and form the ${\mathbb{C}}$-vector space $W:=\bigoplus_{\Omega_{I}}S_{u}$.

• •

For the vector space $I_{W}=I\cap W\subseteq W$, compute

• –

a ${\mathbb{C}}$-basis $(h_{1},\ldots,h_{l})$ and

• –

a description $I_{W}=V(\ell_{1},\ldots,\ell_{m})$ with linear forms $\ell_{i}\in W^{*}$.

• •

With the ${\rm GL}(n)$-action from Construction 2.2 and $Y=(Y_{ij})$, we obtain the ideal

[TABLE]

Output: the ideal $J+J^{\prime}\subseteq S^{\prime}$. Then $V(J+J^{\prime})\subseteq{\rm GL}(n)$ is an algebraic subgroup isomorphic to ${\rm Aut}_{K}(R)$.

Remark 2.7**.**

1. (i)

Algorithms 2.6 and 2.3 do not make use of Gröbner basis computations. However, in Singular, it usually is quicker to compute $J\cap J_{B}$ instead of $J\cdot J_{B}$. 2. (ii)

Computing $G:={\rm Aut}_{K}(R)\subseteq{\rm GL}(n)$ with Algorithm 2.6 enables us to directly compute the number of irreducible components $[G:G^{0}]$ and the dimension of $G$ by Gröbner basis computations.

Example 2.8** (autgradalg.lib III).**

Continuing Example 2.5, let us compute ${\rm Aut}_{K}(R)$. We first switch back to $S$, enter the defining ideal $I$ for $R=S/I$ and start the computation of ${\rm Aut}_{K}(R)$:

setring S; 2. >

ideal I = T(1)*T(6) + T(2)*T(5) + T(3)*T(4) + T(7)*T(8); 3. >

def Sres = autGradAlg(I, TOR); 4. >

setring Sres;

The resulting ring Sres is identical to Sprime. A list stabExported is being exported; the interpretation of the entries is identical to that of the list listAutKS from Example 2.5 with the difference, that the ideal part now contains additional equations describing the stabilizer: for example

stabExported[2][3]

[TABLE]

Moreover, an ideal Jexported is being exported that is the product over all $J_{B}$ as before. Then Sres modulo Jexported is isomorphic to ${\rm Aut}_{K}(R)$. The grading is obtained as before with getVariableWeights().

3. Application: Mori Dream Spaces

In this section, we shortly recall from [10] how the algorithms from the last section can can be applied to a class of varieties in algebraic geometry.

To a normal algebraic variety $X$ over ${\mathbb{C}}$ with finitely generated class group $\operatorname{Cl}(X)$ one can assign a $\operatorname{Cl}(X)$-graded ${\mathbb{C}}$-algebra, its so-called Cox ring

[TABLE]

see e.g. [1] for details on this theory. If $X$ is finitely generated, $X$ is called a Mori dream space. For example, each toric variety or each smooth Fano variety is a Mori dream space [6, 4]. The Cox ring has strong implications on the underlying Mori dream space. More precisely, $X$ can be recovered as a good quotient

[TABLE]

of an open subset $\widehat{X}$ by the characteristic quasitorus $H:={\rm Spec}({\mathbb{C}}[K])$. In fact, $\widehat{X}$ is determined by an ample class $w\in\operatorname{Cl}(X)$. This opens up a computer algebra based approach [9, 12] to Mori dream spaces. In [2], it has been shown that (17) translates to automorphisms of $X$ as follows:

[TABLE]

Here, by ${\rm Aut}_{H}(Y)$ we mean the group of $H$-equivariant automorphisms of $Y$; these are pairs $(\varphi,\psi)$ with $\varphi\colon Y\to Y$ being an automorphism of varieties and $\psi\colon H\to H$ an automorphism of affine algebraic groups such that $\varphi(h\cdot y)=\psi(h)\cdot y$ holds for all $h\in H$ and $y\in Y$. By (20), we directly can compute ${\rm Aut}_{H}(\overline{X})$ with Algorithm 2.6. In the next proposition, we investigate the symmetries of the list of Fano varieties [3].

Proposition 3.1**.**

Let $X_{i}$ be the non-toric terminal Fano threefold of Picard number one with an effective two-torus action from the classification [3, Theorem 1.1].

1. (i)

For all $1\leq i\leq 41$, Algorithm 2.6 is able to compute a presentation of $G_{i}:={\rm Aut}_{H}(\overline{X}_{i})$ as an affine algebraic subgroup $V(J_{i})\subseteq{\rm GL}(n_{i})$. 2. (ii)

Using (i), we list the dimensions $\dim(G_{i})$ and the number of components $[G_{i}:G_{i}^{0}]$ of the following $G_{i}\subseteq{\rm GL}(n_{i})$:

Proof.

This is an application of Algorithm 2.6 and of the Singular commands to compute dimension and absolute components, see for example [8]. We performed the computations on an older machine (Intel celeron CPU, 4 GB Ram) and cancelled them after several seconds. The files are available at [13]. ∎

In [10], the authors have also presented algorithms to compute ${\rm Aut}_{H}(\widehat{X})$ and generators for the Hopf algebra $\mathcal{O}({\rm Aut}(X))$. Both algorithms are also implemented in our library. However, the case $\mathcal{O}({\rm Aut}(X))$ involves a Hilbert basis computation that usually renders the computation infeasible. We therefore finish this note with an example.

Example 3.2** (autgradalg.lib IV).**

In Example 2.8, the algebra $R$ is the Cox ring of a Mori dream space: fix an ample class, say $w:=(0,0,2)\in K\otimes{\mathbb{Q}}$, then $R$ and $w$ define a Mori dream space $X=X(R,w)$. The characteristic quasitorus is $H=({\mathbb{C}}^{*})^{3}\times\{\pm 1\}$.

In 2.8, we have already computed ${\rm Aut}_{H}(\overline{X})\cong G:={\rm Aut}_{K}(R)$. From it, we obtain ${\rm Aut}_{H}(\widehat{X})$ as follows: first, $w$ defines a certain polyhedral cone, the GIT-cone $\lambda(w)$. Then ${\rm Aut}_{H}(\widehat{X})$ is obtained from $G$ by choosing only those elements $(A_{B},B,J_{B})$ of the list stabExported where $B\in{\rm Aut}(\Omega_{S})$ fixes $\lambda(w)$. In our library, you can compute it with (making use of gitfan.lib [5])

intvec w = 1,9,16,0; // drawn in blue 2. >

setring R; // from before 3. >

def RR = autXhat(I, w, TOR); 4. >

setring RR;

w$$q_{1}$$q_{2}$$q_{3}$$q_{4}$$q_{5}$$q_{6}$$q_{7}$$q_{8}$$\lambda(w)

Then a list RES will be exported; it is identical to the list stabExported from Example 2.8 with the difference, that it contains only the element stabExported[1] as the other matrices $B$ do not fix $\lambda(w)$. The computation of generators for $\mathcal{O}({\rm Aut}(X))$ is not feasible here; in priciple, the command is autX(I, w, TOR).