isComplete(NormalToricVariety) -- whether a toric variety is complete

Synopsis

Function: isComplete
Usage:

isComplete X
Inputs:
- X, a normal toric variety,
Outputs:
- a Boolean value, that is true if the normal toric variety is complete

Description

A normal toric variety is complete if any of the following equivalent conditions hold:

the associated complex variety is compact in its classical topology,
the constant map from the normal toric variety to space consisting of a single point is proper,
every one-parameter subgroup of the torus has a limit in the toric variety,
the union of all the cones in the associated fan equals the entire vector space containing it,
every torus-invariant curve lying in the normal toric variety is projective.

For more information, see Theorem 3.4.1 in Cox-Little-Schenck's Toric Varieties.

Affine varieties are not complete.

i1 : AA1 = affineSpace 1

o1 = AA1

o1 : NormalToricVariety

i2 : assert (not isComplete AA1 and isSmooth AA1 and # max AA1 === 1)

i3 : AA3 = affineSpace 3

o3 = AA3

o3 : NormalToricVariety

i4 : assert (not isComplete AA3 and isSmooth AA3 and # max AA3 === 1)

i5 : U = normalToricVariety ({{4,-1,0},{0,1,0}},{{0,1}});

i6 : assert (not isComplete U and isDegenerate U and # max U === 1)

i7 : Q = normalToricVariety ({{1,0,0},{0,1,0},{0,0,1},{1,1,-1}},{{0,1,2,3}})

o7 = Q

o7 : NormalToricVariety

i8 : assert (not isComplete Q and not isSmooth Q and # max Q === 1)

Projective varieties are complete.

i9 : PP1 = toricProjectiveSpace 1;

i10 : assert (isComplete PP1 and isProjective PP1 and isSmooth PP1)

i11 : FF7 = hirzebruchSurface 7;

i12 : assert (isComplete FF7 and isProjective FF7 and isSmooth FF7 and not isFano FF7)

i13 : X = smoothFanoToricVariety (4,120);

i14 : assert (isComplete X and isProjective X and isSmooth X and isFano X)

i15 : P12234 = weightedProjectiveSpace {1,2,2,3,4};

i16 : assert (isComplete P12234 and isProjective P12234 and not isSmooth P12234 and isSimplicial P12234)

i17 : Y = normalToricVariety ( id_(ZZ^3) | - id_(ZZ^3));

i18 : assert (isComplete Y and isProjective Y and not isSmooth Y and not isSimplicial Y)

There are also complete non-projective normal toric varieties.

i19 : X1 = normalToricVariety ({{1,0,0},{0,1,0},{0,0,1},{0,-1,-1},{-1,0,-1},{-2,-1,0}},{{0,1,2},{0,1,3},{1,3,4},{1,2,4},{2,4,5},{0,2,5},{0,3,5},{3,4,5}});

i20 : assert (isComplete X1 and not isProjective X1 and not isSmooth X1 and isWellDefined X1)

i21 : X2 = normalToricVariety ({{1,0,0},{0,1,0},{0,0,1},{0,-1,2},{0,0,-1},{-1,1,-1},{-1,0,-1},{-1,-1,0}},{{0,1,2},{0,2,3},{0,3,4},{0,4,5},{0,1,5},{1,2,7},{2,3,7},{3,4,7},{4,5,6},{4,6,7},{5,6,7},{1,5,7}});

i22 : assert (isComplete X2 and not isProjective X2 and isSmooth X2 and isWellDefined X2)

i23 : X3 = normalToricVariety ({{-1,2,0},{0,-1,0},{1,-1,0},{-1,0,-1},{0,0,-1},{0,1,0},{0,0,1},{1,0,-2}},{{0,1,3},{1,2,3},{2,3,4},{3,4,5},{0,3,5},{0,5,6},{0,1,6},{1,2,6},{2,4,7},{4,5,7},{2,6,7},{5,6,7}});

i24 : assert (isComplete X3 and not isProjective X3 and isSmooth X3 and isWellDefined X3)

To avoid repeating a computation, the package caches the result in the CacheTable of the normal toric variety.

The nonprojective examples are taken from Osamu Fujino and Sam Payne's "Smooth complete toric threefolds with no nontrivial nef line bundles", Japan Academy. Proceedings, Series A, Mathematical Sciences, 81 (2005) 174-179, arXiv:math/0510679.

Ways to use this method: