Introduction

Toric varieties are special instances of schemes. As such, all scheme functionality is available to toric varieties.

Content

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:

underlying_scheme — Method

underlying_scheme(X::AffineNormalToricVariety)

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.

Examples

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)

source

underlying_scheme — Method

underlying_scheme(X::NormalToricVariety)

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.

Examples

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]

source

Contact

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.