# Algorithms for embedded monoids and base point free problems

## TL;DR

This paper introduces algorithms for computations with monoids in finitely generated abelian groups, enabling tests for base point freeness of divisors and related properties in Mori dream spaces.

## Contribution

It provides new algorithms for monoid computations and applies them to problems in algebraic geometry, specifically Mori dream spaces.

## Key findings

- Algorithms for monoid membership testing
- Methods to compute the monoid of base point free divisor classes
- Tools to verify Fujita's base point free conjecture

## Abstract

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 Mori dream space with known canonical class fulfills Fujita's base point free conjecture or not.