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NormalToricVarieties :: isProjective

isProjective -- whether a toric variety is projective

Synopsis

Description

A variety is projective if it can be realized as a closed subvariety of some projective space. For an normal toric variety, this is equivalent to saying that the associated fan is the normal fan of a polytope.

Nontrivial affine varieties are not projective.

i1 : isProjective affineSpace 1

o1 = false
i2 : isProjective affineSpace 3

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

o4 = false
Many of our favour toric varieties are projective.
i5 : isProjective projectiveSpace 1

o5 = true
i6 : isProjective projectiveSpace 3

o6 = true
i7 : isProjective hirzebruchSurface 7

o7 = true
i8 : isProjective smoothFanoToricVariety(3,3)

o8 = true
i9 : X = normalToricVariety(id_(ZZ^3) | -id_(ZZ^3));
i10 : isProjective X

o10 = true
There are complete non-projective normal toric varieties.
i11 : Rho = {{-1,-1,1},{3,-1,1},{0,0,1},{1,0,1},{0,1,1},{-1,3,1},{0,0,-1}};
i12 : 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}}

o12 = {{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}}

o12 : List
i13 : Y = normalToricVariety(Rho,Sigma);
i14 : isComplete Y

o14 = true
i15 : isProjective Y

o15 = true
To determine if a normal toric variety is projective, we use the Gale dual vector configuration associated to the rays; see Theorem V.4.8 in Ewald's book Combinatorial convexity and algebraic geometry for more informaiton.

See also

Ways to use isProjective :