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We propose an algorithm to compute the GIT-fan for torus actions on affinevarieties with symmetries. The algorithm combines computational techniques fromcommutative algebra, convex geometry and group theory. We have implemented ouralgorithm in the Singular library gitfan.lib. Using our implementation, wecompute the Mori chamber decomposition of the cone of movable divisors of$\bar{M}_{0,6}$.

Computing GIT-fans with symmetry and the Mori chamber decomposition of $\bar{M}_{0,6}$