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

Synopsis

• Usage:
normalToricVariety (rayList, coneList)
• Function: normalToricVariety
• Inputs:
• rayList, a list, of lists of integers; each entry is the minimal nonzero lattice point on a ray in the fan
• coneList, a list, of lists of nonnegative integers; each entry indexes the rays defining a maximal cone in the fan
• Optional inputs:
• CoefficientRing => a ring, default value QQ, that determines the coefficient ring of the total coordinate ring
• MinimalGenerators => , default value false, that specifies whether to compute minimal generators
• Variable => , default value x, that specifies the base symbol for the indexed variables in the total coordinate ring
• WeilToClass => , default value null, that specifies the map from the group of torus-invariant Weil divisors to the class group
• Outputs:
• , the normal toric variety determined by the fan

Description

This is the general method for constructing a normal toric variety.

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 d with underlying lattice N = ℤd. 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,…,n. Using this indexing, a maximal cone in the fan corresponds to a sublist of {0,1,…,n}. All maximal cones are listed in coneList.

The first example is projective 2-space 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 = normalToricVariety((({{1, 0}, {0, 1}, {-1, 1}, {-1, 0}, {0, -1}}(,({{0, 1}, {0, 4}, {1, 2}, {2, 3}, {3, 4}} ))))) 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```

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

 `i7 : PP4 = toricProjectiveSpace 4;` ```i8 : rays PP4 o8 = {{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, ------------------------------------------------------------------------ 1}} o8 : List``` ```i9 : max PP4 o9 = {{0, 1, 2, 3}, {0, 1, 2, 4}, {0, 1, 3, 4}, {0, 2, 3, 4}, {1, 2, 3, 4}} o9 : List``` ```i10 : dim PP4 o10 = 4``` ```i11 : ring PP4 o11 = QQ[x , x , x , x , x ] 0 1 2 3 4 o11 : PolynomialRing``` ```i12 : PP4' = normalToricVariety (rays PP4, max PP4, CoefficientRing => ZZ/32003, Variable => y) o12 = normalToricVariety((({{-1, -1, -1, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}(,({{0, 1, 2, 3}, {0, 1, 2, 4}, {0, 1, 3, 4}, {0, 2, 3, 4}, {1, 2, 3, 4}} ))))) o12 : NormalToricVariety``` ```i13 : ring PP4' ZZ o13 = -----[y , y , y , y , y ] 32003 0 1 2 3 4 o13 : PolynomialRing```

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 toric variety. For example, we can choose the opposite generator for the class group of projective space as follows.

 `i14 : PP2 = toricProjectiveSpace 2;` ```i15 : A = fromWDivToCl PP2 o15 = | 1 1 1 | 1 3 o15 : Matrix ZZ <--- ZZ``` ```i16 : source A == weilDivisorGroup PP2 o16 = true``` ```i17 : target A == classGroup PP2 o17 = true``` ```i18 : degrees ring PP2 o18 = {{1}, {1}, {1}} o18 : List``` ```i19 : deg = matrix {toList (3:-1)} o19 = | -1 -1 -1 | 1 3 o19 : Matrix ZZ <--- ZZ``` `i20 : X = normalToricVariety (rays PP2, max PP2, WeilToClass => deg);` ```i21 : A' = fromWDivToCl X o21 = | -1 -1 -1 | 1 3 o21 : Matrix ZZ <--- ZZ``` ```i22 : source A' == weilDivisorGroup X o22 = true``` ```i23 : target A' == classGroup X o23 = true``` ```i24 : degrees ring X o24 = {{-1}, {-1}, {-1}} o24 : List``` ```i25 : (matrix A')* (matrix rays X) o25 = 0 1 2 o25 : Matrix ZZ <--- ZZ```

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.

 `i26 : rayList = {{1,0},{0,1},{-1,1},{-1,0},{0,-1}};` `i27 : coneList = {{0,1},{1,2},{2,3},{3,4},{0,4}};` `i28 : Y = normalToricVariety (rayList, coneList);` ```i29 : fromWDivToCl Y o29 = | 1 -1 1 0 0 | | 1 0 0 1 0 | | 0 1 0 0 1 | 3 5 o29 : Matrix ZZ <--- ZZ``` ```i30 : nefGenerators Y o30 = | 1 0 0 | | 1 0 1 | | 0 1 1 | 3 3 o30 : Matrix ZZ <--- ZZ``` ```i31 : deg = matrix{{1,-1,1,0,0},{0,1,-1,1,0},{0,0,1,-1,1}} o31 = | 1 -1 1 0 0 | | 0 1 -1 1 0 | | 0 0 1 -1 1 | 3 5 o31 : Matrix ZZ <--- ZZ``` `i32 : Y' = normalToricVariety (rays Y, max Y, WeilToClass => deg);` ```i33 : fromWDivToCl Y' o33 = | 1 -1 1 0 0 | | 0 1 -1 1 0 | | 0 0 1 -1 1 | 3 5 o33 : Matrix ZZ <--- ZZ``` ```i34 : nefGenerators Y' o34 = | 1 0 0 | | 0 1 0 | | 0 0 1 | 3 3 o34 : Matrix ZZ <--- ZZ```

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).