Toric varieties are special instances of schemes. As such, all scheme functionality is available to toric varieties.
We aim for a seamless transition among toric varieties and covered schemes.
One advantage is that we can hope for improved performance of scheme functionality by using toric backends when applicable. In addition, one can apply powerful scheme computations to toric settings, thus extending the available toolkit significantly.
The user can extract the scheme corresponding to a toric variety as follows:
For an affine toric scheme $X$, this returns the underlying scheme. In other words, by applying this method, you obtain a scheme that has forgotten its toric origin.
julia> C = positive_hull([1 0; 0 1]) Polyhedral cone in ambient dimension 2 julia> antv = affine_normal_toric_variety(C) Normal, affine toric variety julia> underlying_scheme(antv) Spectrum of quotient of multivariate polynomial ring in 2 variables over QQ by ideal(0)
For a toric covered scheme $X$, this returns the underlying scheme. In other words, by applying this method, you obtain a scheme that has forgotten its toric origin.
julia> P2 = projective_space(NormalToricVariety, 2) Normal, non-affine, smooth, projective, gorenstein, fano, 2-dimensional toric variety without torusfactor julia> underlying_scheme(P2) Scheme over rational field with default covering described by patches 1: normal, affine toric variety 2: normal, affine toric variety 3: normal, affine toric variety in the coordinate(s) 1: [x_1_1, x_2_1] 2: [x_1_2, x_2_2] 3: [x_1_3, x_2_3]
Please direct questions about this part of OSCAR to the following people:
You can ask questions in the OSCAR Slack.
Alternatively, you can raise an issue on github.