A torus is a connected, affine algebraic group that decomposes into a direct product of the multiplicative group over the algebraic closure of the base field. Toric varieties are special varieties with the algebraic action of a torus that have a cellular decomposition into orbits under this action. They can be represented combinatorially as fans, which are unions of rational polyhedral cones in a finite-dimensional rational vector space, or dually, as polytopes in the dual space. Toric-invariant divisors and other interesting geometric structures attached to the variety can be similarly represented.
Toric varieties are an important area of current research. Many interesting rational varieties are toric, including higher-degree Del Pezzo surfaces and many Fano varieties, which are higher-dimensional analogues of Del Pezzo surfaces. The one-to-one correspondence with fans allows the construction of toric varieties and the investigation of their geometric properties through combinatorial geometric methods.
Magma contains a large body of code for toric geometry and working more generally with polytopes and polyhedra. Related to this are a number of important databases of geometric families that can be described in toric terms. In addition to the standard features of a computational toric package, there is a new algorithm to deal with more general maps than the usual toric type, code for Cox rings, and links to Magma schemes.
Along with many other routines for cones and polyhedra, important functions for toric geometry include the following:
The oldest database listed here is not a ‘toric’ database but is similar in style. It is the Magma database of families of K3 surfaces embedded in small-dimensional weighted projective spaces. The computation of these is part of a project that has been running since the mid-1970s, and now contains 24099 entries. The entries are candidates for the Hilbert series of the graded coordinate ring of a K3 surface in weighted projective space.
The toric databases contain data for polytopes that represent toric Fano varieties of certain types. Fano varieties are higher-dimensional analogues of Del Pezzo surfaces: complete varieties whose anticanonical divisor is ample. The three big toric databases contain: