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 *GrÃ¶bner basis and convex polytopes*, University Lecture Series 8. American Mathematical Society, Providence, RI, 1996.

- working with toric maps -- information about toric maps and the induced operations
- inducedMap(ToricMap) -- make the induced map between total coordinate rings (a.k.a. Cox rings)
- ring(NormalToricVariety) -- make the total coordinate ring (a.k.a. Cox ring)