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A software package for Mori dream spaces. (English) Zbl 1348.14002
The article under review gives a description of a software package MDSpackage implemented by the authors of the article for Maple. A Mori dream space $$X$$ is a normal complete variety defined over a field of characteristic 0 which has a finitely generated Cox ring. Such a variety can be encoded by its Cox ring $$\mathcal{R}(X)$$ and some combinatorial data determining a linearization of a certain torus action of $$\mathrm{Spec }\mathcal{R}(X)$$. Many important geometric properties of $$X$$ can be read out from such a description by computations. The MDSpackage package allows, in particular, to obtain information on the divisor class groups of $$X$$, the cone of movable divisors $$\mathrm{Mov}(X)$$ and its chamber decomposition, the ambient toric variety of $$X$$, singularities of $$X$$, and several other aspects under additional assumptions: $$\mathcal{R}(X)$$ being a complete intersection ring or existence of a torus action of complexity one on $$X$$.
In the first part of the article the functions of the package are presented on three examples: a Fano variety, a Gorenstein del Pezzo surface of Picard number one, and a variety with a torus action of complexity one. The second part concerns a single example of a 4-dimensional non-compact symplectic singular variety $$\mathbb{C}^4/G$$, a quotient of $$\mathbb{C}^4$$ by a finite matrix group $$G$$ of order 32. Symplectic resolutions of this quotient singularity were constructed by M. Donten-Bury and J.A. Wiśniewski in [“81 resolutions of a 4-dimensional quotient by a group of order 32”, to appear in Kyoto J. Math.] using methods related to Cox rings. The authors of MDSpackage demonstrate how the Cox ring of all symplectic resolutions of $$\mathbb{C}^4/G$$ can be computed using their software.

##### MSC:
 14-04 Software, source code, etc. for problems pertaining to algebraic geometry 14Q15 Computational aspects of higher-dimensional varieties 14Q10 Computational aspects of algebraic surfaces 14C20 Divisors, linear systems, invertible sheaves 14E30 Minimal model program (Mori theory, extremal rays) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14J45 Fano varieties
##### Keywords:
Mori dream space; Cox ring; computer algebra system
##### Software:
Magma; ToricVarieties; Maple; polymake_toric; MDSpackage; TorDiv; toric; polymake; Convex
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