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Toric Varieties
New Features:
- Magma V2.16 contains the first stage of a large new package for toric
geometry being developed by Gavin Brown, Jaroslaw Buczynski and Alexander
Kasprzyk. It incorporates both the combinatorial and Cox ring approaches.
- The package includes code for cones, fans and polytopes in a rational
vector space. Standard operations and constructions are provided including
the definition of structures, duality, and lattice point counting within
finite polytopes.
- Toric varieties are defined via fans. Combinatorial tests are provided
for the usual geometric properties of the toric variety: singularity,
completeness, projectivity. Standard fan-based constructions such as
weighted blow-ups of toric subsets are included.
- Support is provided for working with torus-invariant divisors and
divisor-class group. This includes arithmetic of divisors, equivalence
tests, computation of the canonical divisor and the construction of
graded cones of the union of Riemann-Roch spaces for all multiples of a
divisor.
- The Cox ring of a toric variety T may be computed. This allows T to be
used as a very general single or multi-graded ambient space. Definition of
arbitrary closed subschemes of T via homogeneous ideals in the Cox ring is
supported. Conversely, Cox rings can be made as abstract objects and
the corresponding toric variety and its combinatorics deduced.
- The basic components of the minimal model program for toric
varieties are incorporated, including extremal contractions, generalised
flips and an explicit tour of the chambers of the mobile cone
(in the sense of Mori dream spaces).
- The package is integrated with the existing Magma scheme structures,
using the Cox ring as the ambient coordinate ring. Many of the basic
scheme operations work for subschemes defined via the Cox ring.
Next: Arithmetic Geometry
Up: Algebraic Geometry
Previous: Algebraic Surfaces
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