### D.15.3 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.

Procedures:

> D.15.3.1 autKS |  | : compute the automorphism group of the basering (must be a polynomial ring) as an algebraic subgroup V(I) of some GL(n)
> D.15.3.2 autGradAlg |  | : compute the automorphism group of R as an algebraic subgroup V(I) of some GL(n).
> D.15.3.3 autGenWeights |  | : compute the automorphisms of the grading group that fix the generator degrees.
> D.15.3.4 stabilizer |  | : compute the stabilizer of the given ideal
> D.15.3.5 autXhat |  | : compute the automorphism group of widehat X as an algebraic subgroup V(I) of some GL(n).
> D.15.3.6 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.Donten-Bury and S.Keicher: 'hilbertBas'.

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

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