7,546 research outputs found
Bunches of cones in the divisor class group -- A new combinatorial language for toric varieties
As an alternative to the description of a toric variety by a fan in the
lattice of one parameter subgroups, we present a new language in terms of what
we call bunches -- these are certain collections of cones in the vector space
of rational divisor classes. The correspondence between these bunches and fans
is based on classical Gale duality. The new combinatorial language allows a
much more natural description of geometric phenomena around divisors of toric
varieties than the usual method by fans does. For example, the numerically
effective cone and the ample cone of a toric variety can be read off
immediately from its bunch. Moreover, the language of bunches appears to be
useful for classification problems.Comment: Minor changes, to appear in Int. Math. Res. No
Three lectures on Cox rings
Notes of an introductory course given at the conference "Torsors: Theory and
Applications" in Edinburgh, January 2011.Comment: Minor corrections, 37 page
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.Comment: 11 pages, minor changes, to appear in LMS Journal of Computation and
Mathematic
- …