Simon Keicher

Computing Cox rings

We consider modifications, for example blow ups, of Mori dream spaces and provide algorithms for investigating the effect on the Cox ring, e.g. testing finite generation or computing an explicit presentation in terms of generators and relations. As a first application, we compute the Cox rings of all Gorenstein log del Pezzo surfaces of Picard number one. Moreover, we show computationally that all smooth rational surfaces of Picard number at most six are Mori dream surfaces and we provide explicit presentations of the Cox ring for those not admitting a torus action. Finally, we provide the Cox rings of projective spaces blown up at a certain special point configurations.

A software package for Mori dream spaces

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 computations. 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/Wi\sniewski.

COMPUTING THE GIT-FAN

We present an algorithm to compute the GIT-fan of algebraic torus actions on affine varieties.

On blowing up the weighted projective plane

We investigate the blow-up of a weighted projective plane at a general point. We provide criteria and algorithms for testing if the result is a Mori dream surface and we compute the Cox ring in several cases. Moreover applications to the study of

A test for monomial containment

\overline{M}_{0,n}

On Chow Quotients of Torus Actions

M¯0,n are discussed.

Computing GIT-fans with symmetry and the Mori chamber decomposition of

Abstract 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.

\bar{M}_{0,6}

Abstract We present an algorithm based on triangular sets to decide whether a given ideal in the polynomial ring contains a monomial.