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NormalToricVarieties :: makeSmooth

makeSmooth -- make a birational smooth toric variety

Synopsis

Description

Every normal toric variety has a resolution of singularities given by another normal toric variety. Given a normal toric variety X, this method makes a new smooth toric variety Y which has a proper birational map to X. The normal toric variety Y is obtained from X by repeatedly blowing up appropriate torus orbit closures (if necessary the makeSimplicial method is also used with the specified strategy). A minimal number of blow-ups are used.

As a simple example, we can resolve a simplicial affine singularity.

U = normalToricVariety({{4,-1},{0,1}},{{0,1}});
isSmooth U
V = makeSmooth U;
isSmooth V
rays V, max V
set rays V - set rays U
There is one additional rays, so only one blowup was needed.

To resolve the singularities of this simplicial projective fourfold, we need eleven blowups.

W = weightedProjectiveSpace {1,2,3,4,5};
dim W
isSimplicial W
isSmooth W
W' = makeSmooth W;
isSmooth W'
R = set rays W' - set rays W
#R
If the initial toric variety is smooth, then this method simply returns it.
AA1 = affineSpace 1;
AA1 === makeSmooth AA1
PP2 = projectiveSpace 2;
PP2 === makeSmooth PP2
In the next example, we resolve the singularities of a non-simplicial projective threefold.
X = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));
isSimplicial X
isSmooth X
X' = makeSmooth X;
isSmooth X'
R = set rays X' - set rays X
#R
We also demonstrate this method on a complete simplicial non-projective threefold.
Rho = {{-1,-1,1},{3,-1,1},{0,0,1},{1,0,1},{0,1,1},{-1,3,1},{0,0,-1}};
Sigma = {{0,1,3},{0,1,6},{0,2,3},{0,2,5},{0,5,6},{1,3,4},{1,4,5},{1,5,6},{2,3,4},{2,4,5}};
Z = normalToricVariety(Rho,Sigma);
isSimplicial Z
isSmooth Z
isComplete Z
isProjective Z
Z' = makeSmooth Z;
isSmooth Z'
R = set rays Z' - set rays Z
#R
We end with a degenerate example.
Y = normalToricVariety({{1,0,0,0},{0,1,0,0},{0,0,1,0},{1,-1,1,0},{1,0,-2,0}},{{0,1,2,3},{0,4}});
isDegenerate Y
Y' = makeSmooth Y;
isSmooth Y'

Caveat

A singular normal toric variety almost never has a unique minimal esolution. This method returns only of one of the many minimal resolutions.

See also

Ways to use makeSmooth :