- Usage:
`ring X`

- Function: ring
- Inputs:
`X`, a normal toric variety

- Outputs:
- a polynomial ring, the total coordinate ring

The total coordinate ring, which is also known as the Cox ring, of a normal toric variety is a polynomial ring in which the variables correspond to the rays in the fan. The map from the group of torus-invarient Weil divisors to the class group endows this ring with a grading by the class group.

The total coordinate ring for projective space is the standard graded polynomial ring.

i1 : PP3 = toricProjectiveSpace 3; |

i2 : S = ring PP3; |

i3 : assert (isPolynomialRing S and isCommutative S) |

i4 : gens S o4 = {x , x , x , x } 0 1 2 3 o4 : List |

i5 : degrees S o5 = {{1}, {1}, {1}, {1}} o5 : List |

i6 : assert (numgens S == #rays PP3) |

i7 : coefficientRing S o7 = QQ o7 : Ring |

For a product of projective spaces, the total coordinate ring has a bigrading.

i8 : X = toricProjectiveSpace(2) ** toricProjectiveSpace(3); |

i9 : gens ring X o9 = {x , x , x , x , x , x , x } 0 1 2 3 4 5 6 o9 : List |

i10 : degrees ring X o10 = {{1, 0}, {1, 0}, {1, 0}, {0, 1}, {0, 1}, {0, 1}, {0, 1}} o10 : List |

A Hirzebruch surface also has a *ℤ ^{2}*-grading.

i11 : FF3 = hirzebruchSurface 3; |

i12 : gens ring FF3 o12 = {x , x , x , x } 0 1 2 3 o12 : List |

i13 : degrees ring FF3 o13 = {{1, 0}, {-3, 1}, {1, 0}, {0, 1}} o13 : List |

The total coordinate ring is not yet implemented when the toric variety is degenerate or the class group has torsion.

- Total coordinate rings and coherent sheaves
- rays(NormalToricVariety) -- get the rays of the associated fan
- classGroup -- make the class group
- WeilToClass -- make a normal toric variety
- fromWDivToCl(NormalToricVariety) -- get the map from the group of Weil divisors to the class group
- ideal(NormalToricVariety) -- make the irrelevant ideal
- sheaf(NormalToricVariety,Module) -- make a coherent sheaf