Algorithms for embedded monoids and base point free problems

We present algorithms for basic computations with monoids in finitely generated abelian groups such as monoid membership testing and computing an element of the conductor ideal. Applying them to Mori dream spaces, we obtain algorithms to test whether a Weil divisor class of a given Mori dream space is base point free, to compute generators of the monoid of base point free Cartier divisor classes and to test whether a Q-factorial Mori dream space with known canonical class fulfills Fujita's base point free conjecture or not.