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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 : AA1 = affineSpace 1

o1 = normalToricVariety((({{1}}(,({{0}} )))))

o1 : NormalToricVariety
i2 : assert (not isComplete AA1 and isSmooth AA1 and # max AA1 === 1)
i3 : AA3 = affineSpace 3

o3 = normalToricVariety((({{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}(,({{0, 1, 2}} )))))

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

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)

Reference

The nonprojective examples are taken from [Osamu Fujino and Sam Payne, 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