The closure of image of a morphism $f : X \to Y$ is a closed subscheme in $Y$. All closed subschemes in normal toric variety $Y$ correspond to a saturated homogeneous ideal in the total coordinate ring (a.k.a. Cox ring) of $Y$. For more information, see Proposition 5.2.4 in Cox-Little-Schenck's Toric Varieties. This method returns the saturated homogeneous ideal corresponding to the closure of the image $f$.
The closure of a distinguished affine open set in the projective space is the entire space.
i1 : A = affineSpace 4; |
i2 : P = toricProjectiveSpace 4; |
i3 : iota = map(P, A, 1)
o3 = | 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
o3 : ToricMap P <--- A
|
i4 : ideal iota
o4 = ideal 0
o4 : Ideal of QQ[x ..x ]
0 4
|
i5 : assert (isWellDefined iota and ideal iota == 0) |
The twisted cubic curve is the image of a map from the projective line to the projective $3$-space.
i6 : X = toricProjectiveSpace 1; |
i7 : Y = toricProjectiveSpace 3; |
i8 : f = map(Y, X, matrix{{1}, {2}, {3}})
o8 = | 1 |
| 2 |
| 3 |
o8 : ToricMap Y <--- X
|
i9 : S = ring Y; |
i10 : I = ideal f
2 2
o10 = ideal (x - x x , x x - x x , x - x x )
2 1 3 1 2 0 3 1 0 2
o10 : Ideal of S
|
i11 : assert (isWellDefined f and isHomogeneous I and
I == saturate(I, ideal Y) and I == ker inducedMap f and
I == minors(2, matrix{{S_0, S_1, S_2}, {S_1, S_2, S_3}}))
|
Thirdly, we have the image of diagonal embedding of the projective $4$-space.
i12 : (Y2 = Y ** Y, R = ring Y2); |
i13 : g = diagonalToricMap(Y, 2); o13 : ToricMap Y2 <--- Y |
i14 : J = ideal g
o14 = ideal (x x - x x , x x - x x , x x - x x , x x - x x , x x - x x ,
3 6 2 7 3 5 1 7 2 5 1 6 3 4 0 7 2 4 0 6
-----------------------------------------------------------------------
x x - x x )
1 4 0 5
o14 : Ideal of R
|
i15 : assert (isWellDefined g and isHomogeneous J and
J == saturate(J, ideal Y2) and
J == minors(2, matrix{{R_0,R_1,R_2,R_3},{R_4,R_5,R_6,R_7}}))
|
The algorithm used is a minor variant of Algorithm 12.3 in Bernd Sturmfels Gröbner basis and convex polytopes, University Lecture Series 8. American Mathematical Society, Providence, RI, 1996.