Computing Cox rings

Abstract

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.

33 pages, minor changes. To appear in Mathematics of Computation

References in corpus (3)

• On Chow quotients of torus actions
• Rees algebras of square-free monomial ideals
• Modular absolute decomposition of equidimensional polynomial ideals

Cited by in corpus (11)

• On blowing up the weighted projective plane
• On torus actions of higher complexity
• On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32
• A software package for Mori dream spaces
• Cox rings of cubic surfaces and Fano threefolds
• Iteration of Cox rings of klt singularities
• Spherical blow-ups of Grassmannians and Mori Dream Spaces
• On the Cox ring of blowing up the diagonal
• On del Pezzo elliptic varieties of degree $\le 4$
• The Picard index of a surface with torus action
• The cone of curves and the Cox ring of rational surfaces over Hirzebruch surfaces