Rational Parametrization of Rational Surfaces

parametrization — Method

parametrization(X::AbsProjectiveVariety)

Given a smooth rational surface X which is linearly normal in the given embedding, return a rational parametrization of X.

Note

The function does not check whether X is smooth. If you are uncertain, enter is_smooth(X) first.

Note

The function does not check rationality. In fact, at current state, OSCAR does not offer a direct check for this.

Note

The function makes use of the adjunction process. It returns an error message if the terminal object of the adjunction process is not the projective plane. See the OSCAR documentation for information on the adjunction process.

Examples

julia> X = bordiga()
[ Info: upgrading serialized data....
Projective variety
  in projective 4-space over GF(31991) with coordinates [x, y, z, u, v]
defined by ideal with 4 generators

julia> dim(X)
2

julia> codim(X)
2

julia> phi = parametrization(X);

julia> domain(phi)
Multivariate polynomial ring in 5 variables over GF(31991) graded by
  x -> [1]
  y -> [1]
  z -> [1]
  u -> [1]
  v -> [1]

julia> codomain(phi)
Multivariate polynomial ring in 3 variables over GF(31991) graded by
  z[1] -> [1]
  z[2] -> [1]
  z[3] -> [1]

julia> [degree(phi(x)) for x in gens(ambient_coordinate_ring(X))]
5-element Vector{FinGenAbGroupElem}:
 [4]
 [4]
 [4]
 [4]
 [4]

source