next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
NormalToricVarieties :: isWellDefined(NormalToricVariety)

isWellDefined(NormalToricVariety) -- whether a toric variety is well-defined

Synopsis

Description

A pair of lists (Rho,Sigma) correspond to a well-defined normal toric variety if the following conditions hold:
  • the union of the elements of Sigma equals the set of indices of elements of Rho
  • no element of Sigma is properly contained in another element of Sigma
  • all elements of Rho have the same length
  • all elements of Rho are lists of integers
  • the rays indexed by an element of Sigma generate a strongly convex cone
  • the rays indexed by an element of Sigma are the unique minimal lattice points for the cone they generate
  • the intersection of the cones associated to two elements of Sigma is a face of each cone.

The first examples illustrate that small projective spaces are well-defined.

for d from 1 to 6 list isWellDefined projectiveSpace d
The second examples show that a randomly selected Kleinschmidt toric variety and a weighted projective space are also well-defined.
setRandomSeed(currentTime());
a = sort apply(3, i -> random(7))
isWellDefined kleinschmidt(4,a)
q = apply(5, j -> random(1,9));
while not all(subsets(q,#q-1), s -> gcd s === 1) do ( q = apply(5, j -> random(1,9)));
q
isWellDefined weightedProjectiveSpace q
The next eight examples illustrate various ways that two lists can fail to define a normal toric variety. By making the current debugging level greater than one, one gets some addition information about the nature of the failure.
Sigma = max projectiveSpace 2;
X1 = normalToricVariety({{-1,-1},{1,0},{0,1},{-1,0}},Sigma);
isWellDefined X1
debugLevel = 1;
isWellDefined X1
Sigma' = {{0,1},{0,3},{1,2},{2,3},{3}};
X2 = normalToricVariety({{-1,0},{0,-1},{1,-1},{0,1}},Sigma');
isWellDefined X2
X3 = normalToricVariety({{-1,-1},{1,0},{0,1,1}},Sigma);
isWellDefined X3
X4 = normalToricVariety({{-1,-1/1},{1,0},{0,1}},Sigma);
isWellDefined X4
X5 = normalToricVariety({{1,0},{0,1},{-1,0}},{{0,1,2}});
isWellDefined X5
X6 = normalToricVariety({{1,0},{0,1},{1,1}},{{0,1,2}});
isWellDefined X6
X7 = normalToricVariety({{1,0,0},{0,1,0},{0,0,2}},{{0,1,2}});
isWellDefined X7
X8 = normalToricVariety({{1,0},{0,1},{1,1}},{{0,1},{1,2}});
isWellDefined X8

See also