
D.15.1 autgradalg_lib
 Library:
 autgradalg.lib
 Purpose:
 Compute automorphism groups of pointedly graded algebras and of Mori dream spaces.
 Authors:
 Simon Keicher
 Overview:
 This library provides a framework for computing automorphisms of integral, finitely generated algebras 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).
 Assumptions:
 * the algebra R is given as factor algebra S/I with a graded polynomial ring S = KK[T_1,...,T_r]. We will always assume that the basering is S and it is given over the rationals QQ or a number field QQ(a).
* R must be minimally presented, i.e., I is contained in <T_1,...,T_r>^2.
* S (and hence R) are graded via 'setBaseMultigrading(Q)' from 'multigrading.lib'. The last rows of the matrix Q are interpreted over ZZ/a_iZZ if the respective entry of the list TOR is a_i and has been provided as parameter to the respective function. (See the functions for more details.)
* For all 1 <= i <= r: I_{w_i} = 0 where w_i := deg(T_i).
* the grading is pointed, i.e., no generator has degree 0 and the cone over all generator degrees is pointed.
* For Mori dream spaces X, we assume them to be given as X = X(R,w) with the Cox ring R of X (given as the algebra R before) and an ample class w in the grading group K with the torsion entries removed.
 Note:
 we require the user to execute 'LIB'gfanlib.so" before using this library.
Procedures:
D.15.1.1 autKS   : compute the automorphism group of the basering (must be a polynomial ring) as an algebraic subgroup V(I) of some GL(n) 
D.15.1.2 autGradAlg   : compute the automorphism group of R as an algebraic subgroup V(I) of some GL(n). 
D.15.1.3 autGenWeights   : compute the automorphisms of the grading group that fix the generator degrees. 
D.15.1.4 stabilizer   : compute the stabilizer of the given ideal 
D.15.1.5 autXhat   : compute the automorphism group of widehat X as an algebraic subgroup V(I) of some GL(n). 
D.15.1.6 autX   : compute the automorphism group of X=X(R,w) as an algebraic subgroup V(I) of some GL(n). 
 Note:
 the following functions were taken from 'quotsingcox.lib' by M.DontenBury and S.Keicher: 'hilbBas'.
