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autgradalg.lib

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.

Table of contents

Introduction

Summary and references

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.

Assumptions

Please refer to the detailed description for the full list of assumptions. They include

  • R is given as factor algebra S/I0 with a graded polynomial ring S 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 component I_{w_i} is trivial where w_i is the degree of the i-th generator of R.
  • For Mori dream spaces X, we assume them to be given as X = X(R,w) with the Cox ring R of X and the free part of an ample class w.

How to use the library

Installation

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.

  1. Either
    • download the file lib/autgradalg.lib and cd to the folder containing it
    • or clone this repository and cd to the lib folder.
  2. Execute Singular and then load the library with the commands
    LIB 'gfanlib.so'
    LIB 'autgradalg.lib'

Usage and examples

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.

Procedures

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 subgroup V(I) of some GL(n).
  • autGradAlg(I0, TOR): compute the automorphism group of the algebra basering/I0 as an algebraic subgroup V(I) of some GL(n).
  • autGenWeights(TOR): compute the automorphisms of the grading group of the basering that fix the generator degrees.
  • stabilizer(I0, TOR): compute the stabilizer of the given ideal I0.
  • autXhat(I0, w, TOR): compute the automorphism group of \widehat X as an algebraic subgroup V(I) of some GL(n).
  • autX(I0, w, TOR): compute the automorphism group of X=X(R,w) as an algebraic subgroup V(I) of some GL(n). Note: usually very expensive.
  • shrink(J): returns a new ring obtained from the old one by removing variables appearing as generators in J.

Testing

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;

Logging

You can increase the loglevel with

printlevel = 2;

Note

This library comes without any warranty whatsoever. Use it at your own risk.

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