autgradalg.lib
is a library for the free computer algebra system Singular to compute automorphism groups of pointedly graded algebras and of Mori dream spaces.
autgradalg.lib
provides a framework for computing automorphisms of integral, finitely generated algebras R
that are graded by a finitely generated abelian group. This library also contains functions to compute automorphism groups of Mori dream spaces. The results are ideals I
such that the respective automorphism group is isomorphic to the subgroup V(I)
in some GL(n)
.
The methods can be used to compute symmetries of homogeneous ideals.
autgradalg.lib
and the necessary background have been described in this paper.
It is an implementation of the algorithms developed in Computing automorphisms of Mori dream spaces.
Please refer to the detailed description for the full list of assumptions. They include
R
is given as factor algebraS/I0
with a graded polynomial ringS
over the rationals.R
must be minimally presented, i.e., I is contained in<T_1,...,T_r>^2
.- the grading is pointed, i.e., no generator has degree
0
and the cone over all generator degrees is pointed. - For all
1 <= i <= r
the homogeneous componentI_{w_i}
is trivial wherew_i
is the degree of the i-th generator ofR
. - For Mori dream spaces
X
, we assume them to be given asX = X(R,w)
with the Cox ringR
ofX
and the free part of an ample classw
.
You need to have Singular installed. The library has been developed for version 4.1.1. We assume it has been installed on a UNIX machine and is available from the command line.
- Either
- download the file
lib/autgradalg.lib
andcd
to the folder containing it - or clone this repository and
cd
to thelib
folder.
- download the file
- Execute
Singular
and then load the library with the commandsLIB 'gfanlib.so' LIB 'autgradalg.lib'
You can find examples in the folder examples
.
To run any file, say FOO.sing
, cd
to the corresponding directory, and run Singular FOO.sing
on the command line.
- the folder
examples/paper
lists files to recompute the examples used in the paper. - the folder
examples/fanos
lists the files to verify the results on Fano varieties listed in the Proposition in the paper.
The following functions are available from our library:
type help FUNCTIONNAME;
to see further information.
Again, more information and examples can be found here.
autKS()
: compute the automorphism group of the basering (must be a polynomial ring) as an algebraic subgroupV(I)
of someGL(n)
.autGradAlg(I0, TOR)
: compute the automorphism group of the algebrabasering/I0
as an algebraic subgroupV(I)
of someGL(n)
.autGenWeights(TOR)
: compute the automorphisms of the grading group of thebasering
that fix the generator degrees.stabilizer(I0, TOR)
: compute the stabilizer of the given idealI0
.autXhat(I0, w, TOR)
: compute the automorphism group of\widehat X
as an algebraic subgroupV(I)
of someGL(n)
.autX(I0, w, TOR)
: compute the automorphism group ofX=X(R,w)
as an algebraic subgroupV(I)
of someGL(n)
. Note: usually very expensive.shrink(J)
: returns a new ring obtained from the old one by removing variables appearing as generators inJ
.
To test all of the previously listed procedures, go to the test
folder, and execute
Singular test.sing
You can test only one of the previously listed procedures, say foo
, by starting Singular
, loading the library and then running
example foo;
You can increase the loglevel with
printlevel = 2;
This library comes without any warranty whatsoever. Use it at your own risk.