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NormalToricVarieties :: NormalToricVariety _ Array

NormalToricVariety _ Array -- make a canonical inclusion into a product

Synopsis

Description

A product of varieties is equipped with canonical inclusion maps from a product of any subset of its factors. Given a product of normal toric varieties and a nonempty array, this methods provides a concise way to make these toric maps.

The product of two normal toric varieties has inclusions from each factor.

i1 : Y0 = toricProjectiveSpace 1;
i2 : Y1 = hirzebruchSurface 3;
i3 : X = Y0 ** Y1;
i4 : X_[0]

o4 = | 1 |
     | 0 |
     | 0 |

o4 : ToricMap X <--- Y0
i5 : assert (isWellDefined X_[0] and source X_[0] === Y0 and target X_[0] === X)
i6 : X_[1]

o6 = | 0 0 |
     | 1 0 |
     | 0 1 |

o6 : ToricMap X <--- Y1
i7 : assert (isWellDefined X_[1] and source X_[1] === Y1 and target X_[1] === X)

The canonical inclusions interact with the canonical projections in the expected way.

i8 : assert (X^[0] * X_[0] == id_Y0 and X^[1] * X_[1] == id_Y1)
i9 : assert (X^[1] * X_[0] == map(Y1, Y0, 0) and X^[0] * X_[1] == map(Y0, Y1, 0))

If A indexes all the factors, then we simply obtain the identity map on X.

i10 : X_[0,1]

o10 = | 1 0 0 |
      | 0 1 0 |
      | 0 0 1 |

o10 : ToricMap X <--- X
i11 : assert (X_[0,1] == id_X)

When there are more than two factors, we also obtain inclusions from any subset of the factors.

i12 : Z = Y0 ^** 3;
i13 : Z_[0]

o13 = | 1 |
      | 0 |
      | 0 |

o13 : ToricMap Z <--- Y0
i14 : Z_[1]

o14 = | 0 |
      | 1 |
      | 0 |

o14 : ToricMap Z <--- Y0
i15 : Z_[2]

o15 = | 0 |
      | 0 |
      | 1 |

o15 : ToricMap Z <--- Y0
i16 : assert all (3, i -> isWellDefined Z_[i] and source Z_[i] === Y0 and target Z_[i] === Z)
i17 : Z_[0,1]

o17 = | 1 0 |
      | 0 1 |
      | 0 0 |

o17 : ToricMap Z <--- normalToricVariety ({{-1, 0}, {1, 0}, {0, -1}, {0, 1}}, {{0, 2}, {0, 3}, {1, 2}, {1, 3}})
i18 : Z_[0,2]

o18 = | 1 0 |
      | 0 0 |
      | 0 1 |

o18 : ToricMap Z <--- normalToricVariety ({{-1, 0}, {1, 0}, {0, -1}, {0, 1}}, {{0, 2}, {0, 3}, {1, 2}, {1, 3}})
i19 : Z_[1,2]

o19 = | 0 0 |
      | 1 0 |
      | 0 1 |

o19 : ToricMap Z <--- normalToricVariety ({{-1, 0}, {1, 0}, {0, -1}, {0, 1}}, {{0, 2}, {0, 3}, {1, 2}, {1, 3}})
i20 : assert (isWellDefined Z_[1,2] and source Z_[1,2] === Y0 ** Y0)
i21 : Z_[0,1,2]

o21 = | 1 0 0 |
      | 0 1 0 |
      | 0 0 1 |

o21 : ToricMap Z <--- Z
i22 : assert (Z_[0,1,2] == id_Z)

When the normal toric variety is not constructed as a product, this method only reproduces the identity map.

i23 : components Y1

o23 = {Y1}

o23 : List
i24 : Y1_[0]

o24 = | 1 0 |
      | 0 1 |

o24 : ToricMap Y1 <--- Y1
i25 : assert (Y1_[0] == id_Y1)

See also