# ideal(ToricMap) -- make the ideal defining the closure of the image

## Synopsis

• Function: ideal
• Usage:
ideal f
• Inputs:
• f, ,
• Outputs:
• an ideal, in the homogeneous coordinate ring (a.k.a. Cox ring) of the target

## Description

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.