We present an algorithm to decide whether a given ideal in the polynomial ring contains a monomial without using Grobner bases, factorization or sub-resultant computations.
Mori dream spaces form a large example class of algebraic varieties, comprising the well known toric varieties. We provide a first software package for the explicit treatment of Mori dream spaces and demonstrate its use by presenting basic sample com
putations. The software package is accompanied by a Cox ring database which delivers defining data for Cox rings and Mori dream spaces in a suitable format. As an application of the package, we determine the common Cox ring for the symplectic resolutions of a certain quotient singularity investigated by Bellamy/Schedler and Donten-Bury/Wisniewski.
We determine the Cox rings of the minimal resolutions of cubic surfaces with at most rational double points, of blow ups of the projective plane at non-general configurations of six points and of three dimensional smooth Fano varieties of Picard numbers one and two.
