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NormalToricVarieties :: normalToricVariety(List,List)

normalToricVariety(List,List) -- make a normal toric variety

Synopsis

Description

This is the main method for constructing a normal toric variety. Almost all other constructors invoke it.

A normal toric variety corresponds to a strongly convex rational polyhedral fan in affine space. In this package, the fan associated to a normal $d$-dimensional toric variety lies in the rational vector space $\QQ^d$ with underlying lattice $N = \ZZ^d$. The fan is encoded by the minimal nonzero lattice points on its rays and the set of rays defining the maximal cones (meaning cones that are not proper subsets of another cone in the fan). More precisely, rayList lists the minimal nonzero lattice points on each ray (a.k.a. one-dimensional cone) in the fan. Each lattice point is a list of integers. The rays are ordered and indexed by nonnegative integers: $0,1,\dots,n$. Using this indexing, a maximal cone in the fan corresponds to a sublist of $\{0,1,\dots,n\}$. All maximal cones are listed in coneList. More information explaining the correspondence between normal toric varieties and strongly convex rational polyhedral fans appears in Subsection 3.1 of Cox-Little-Schenck.

The first example is projective plane blown up at two points.

i1 : rayList = {{1,0},{0,1},{-1,1},{-1,0},{0,-1}}

o1 = {{1, 0}, {0, 1}, {-1, 1}, {-1, 0}, {0, -1}}

o1 : List
i2 : coneList = {{0,1},{1,2},{2,3},{3,4},{0,4}}

o2 = {{0, 1}, {1, 2}, {2, 3}, {3, 4}, {0, 4}}

o2 : List
i3 : X = normalToricVariety (rayList, coneList)

o3 = X

o3 : NormalToricVariety
i4 : rays X

o4 = {{1, 0}, {0, 1}, {-1, 1}, {-1, 0}, {0, -1}}

o4 : List
i5 : max X

o5 = {{0, 1}, {0, 4}, {1, 2}, {2, 3}, {3, 4}}

o5 : List
i6 : dim X

o6 = 2
i7 : assert (isWellDefined X and isProjective X and isSmooth X)

The second example illustrates the data defining projective $4$-space.

i8 : PP4 = toricProjectiveSpace 4;
i9 : rays PP4

o9 = {{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0,
     ------------------------------------------------------------------------
     1}}

o9 : List
i10 : max PP4

o10 = {{0, 1, 2, 3}, {0, 1, 2, 4}, {0, 1, 3, 4}, {0, 2, 3, 4}, {1, 2, 3, 4}}

o10 : List
i11 : dim PP4

o11 = 4
i12 : ring PP4

o12 = QQ[x ..x ]
          0   4

o12 : PolynomialRing
i13 : PP4' = normalToricVariety (rays PP4, max PP4, CoefficientRing => ZZ/32003, Variable => y)

o13 = PP4'

o13 : NormalToricVariety
i14 : ring PP4'

        ZZ
o14 = -----[y ..y ]
      32003  0   4

o14 : PolynomialRing
i15 : assert (isWellDefined PP4 and isProjective PP4 and isSmooth PP4)

The optional argument WeilToClass allows one to specify the map from the group of torus-invariant Weil divisors to the class group. In particular, this allows the user to choose her favourite basis for the class group. This map also determines the grading on the total coordinate ring of the normal toric variety. For example, we can choose the opposite generator for the class group of projective space as follows.

i16 : PP2 = toricProjectiveSpace 2;
i17 : A = fromWDivToCl PP2

o17 = | 1 1 1 |

               1        3
o17 : Matrix ZZ  <--- ZZ
i18 : source A == weilDivisorGroup PP2

o18 = true
i19 : target A == classGroup PP2

o19 = true
i20 : degrees ring PP2

o20 = {{1}, {1}, {1}}

o20 : List
i21 : deg = matrix {toList (3:-1)}

o21 = | -1 -1 -1 |

               1        3
o21 : Matrix ZZ  <--- ZZ
i22 : X = normalToricVariety (rays PP2, max PP2, WeilToClass => deg);
i23 : A' = fromWDivToCl X

o23 = | -1 -1 -1 |

               1        3
o23 : Matrix ZZ  <--- ZZ
i24 : source A' == weilDivisorGroup X

o24 = true
i25 : target A' == classGroup X

o25 = true
i26 : degrees ring X

o26 = {{-1}, {-1}, {-1}}

o26 : List
i27 : (matrix A') * (matrix rays X)

o27 = 0

               1        2
o27 : Matrix ZZ  <--- ZZ
i28 : assert (isWellDefined X and isProjective X and isSmooth X)

The integer matrix A should span the kernel of the matrix whose columns are the minimal nonzero lattice points on the rays of the fan.

We can also choose a basis for the class group of a blow-up of the projective plane such that the nef cone is the positive quadrant.

i29 : rayList = {{1,0},{0,1},{-1,1},{-1,0},{0,-1}};
i30 : coneList = {{0,1},{1,2},{2,3},{3,4},{0,4}};
i31 : Y = normalToricVariety (rayList, coneList);
i32 : fromWDivToCl Y

o32 = | 1 -1 1 0 0 |
      | 1 0  0 1 0 |
      | 0 1  0 0 1 |

               3        5
o32 : Matrix ZZ  <--- ZZ
i33 : nefGenerators Y

o33 = | 1 0 0 |
      | 1 0 1 |
      | 0 1 1 |

               3        3
o33 : Matrix ZZ  <--- ZZ
i34 : deg = matrix{{1,-1,1,0,0},{0,1,-1,1,0},{0,0,1,-1,1}}

o34 = | 1 -1 1  0  0 |
      | 0 1  -1 1  0 |
      | 0 0  1  -1 1 |

               3        5
o34 : Matrix ZZ  <--- ZZ
i35 : Y' = normalToricVariety (rays Y, max Y, WeilToClass => deg);
i36 : fromWDivToCl Y'

o36 = | 1 -1 1  0  0 |
      | 0 1  -1 1  0 |
      | 0 0  1  -1 1 |

               3        5
o36 : Matrix ZZ  <--- ZZ
i37 : nefGenerators Y'

o37 = | 1 0 0 |
      | 0 1 0 |
      | 0 0 1 |

               3        3
o37 : Matrix ZZ  <--- ZZ
i38 : assert (isWellDefined Y and isWellDefined Y')

Caveat

This method assumes that the lists rayList and coneList correctly encode a strongly convex rational polyhedral fan. One can verify this by using isWellDefined(NormalToricVariety).

See also