I have two curves in a two-dimensional projective space. I compute the scheme theoretic-intersection, and then take the irreducible components. Each of these things really is an honest point in P2. How can I get from the closed subscheme form of these points to the actual coordinates?
Here's an example. I compute the intersection of the (projective closures of) y=0 and y=x^2-1. The result should be two points, namely [1,0,1] and [-1,0,1].
P2 = ProjectiveSpace(QQ,2,'x,y,z')
P2.inject_variables()
V1 = P2.subscheme(y)
V2 = P2.subscheme(y*z-x^2+z^2)
W = V1.intersection(V2)
[ p1, p2 ] = W.irreducible_components()
I can only get as far as p1 and p2, which Sage thinks of as
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
y,
x - z,
How to I turn it into a point in projective space, like P2([1,0,1])?
