Rational Parametrization of Rational Surfaces
What we present here relies on the function adjunction_process discussed in the previous section.
Parametrization
parametrization — Methodparametrization(X::AbsProjectiveVariety)Given a smooth rational surface X which is linearly normal in the given embedding, return a rational parametrization of X.
The function does not check whether X is smooth. If you are uncertain, enter is_smooth(X) first.
The function does not check rationality. In fact, at current state, OSCAR does not offer a direct check for this.
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()
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]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.