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NormalToricVarieties :: isComplete(NormalToricVariety)

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

Synopsis

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.

Affine varieties are not complete.

i1 : isComplete affineSpace 1

o1 = false
i2 : isComplete affineSpace 3

o2 = false
i3 : U = normalToricVariety({{4,-1,0},{0,1,0}},{{0,1}});
i4 : isComplete U

o4 = false
Projective varieties are complete.
i5 : isComplete projectiveSpace 1

o5 = true
i6 : isComplete projectiveSpace 3

o6 = true
i7 : isComplete hirzebruchSurface 7

o7 = true
There are also complete non-projective normal toric varieties.
i8 : R1 = {{1,0,0},{0,1,0},{0,0,1},{0,-1,-1},{-1,0,-1},{-2,-1,0}};
i9 : S1 = {{0,1,2},{0,1,3},{1,3,4},{1,2,4},{2,4,5},{0,2,5},{0,3,5},{3,4,5}};
i10 : X1 = normalToricVariety(R1,S1);
i11 : isComplete X1

o11 = true
i12 : isProjective X1

o12 = false
i13 : isSmooth X1

o13 = false
i14 : R2 = {{1,0,0},{0,1,0},{0,0,1},{0,-1,2},{0,0,-1},{-1,1,-1},{-1,0,-1},{-1,-1,0}};
i15 : S2 = {{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}};  
i16 : X2 = normalToricVariety(R2,S2);    
i17 : isComplete X2

o17 = true
i18 : isProjective X2 

o18 = false
i19 : isSmooth X2

o19 = true
i20 : R3 = {{-1,2,0},{0,-1,0},{1,-1,0},{-1,0,-1},{0,0,-1},{0,1,0},{0,0,1},{1,0,-2}};
i21 : S3 = {{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}};
i22 : X3 = normalToricVariety(R3,S3);    
i23 : isComplete X3

o23 = true
i24 : isProjective X3 

o24 = false
i25 : isSmooth X3    

o25 = true
The nonprojective examples are taken from Osamu Fujino and Sam Payne's paper Smooth complete toric threefolds with non nontrivial nef line bundles Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 10, 174--179.

See also