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NormalToricVarieties :: vertices(ToricDivisor)

vertices(ToricDivisor) -- compute the vertices of the associated polytope

Synopsis

Description

On a complete normal toric variety, the polyhedron associated to a Cartier divisor is a lattice polytope. Given a torus-invariant Cartier divisor on a normal toric variety, this method returns an integer matrix whose columns correspond to the vertices of the associated lattice polytope. For a non-effective Cartier divisor, this methods returns null. When the divisor is ample, the normal fan the corresponding polytope equals the fan associated to the normal toric variety.

On the projective plane, the associate polytope is either empty, a point, or a triangle.

i1 : PP2 = projectiveSpace 2;
i2 : vertices (-PP2_0)
i3 : null === vertices (- PP2_0)

o3 = true
i4 : vertices (0*PP2_0)

o4 = 0

              2        1
o4 : Matrix ZZ  <--- ZZ
i5 : isAmple PP2_0

o5 = true
i6 : V1 = vertices (PP2_0)

o6 = | 0 1 0 |
     | 0 0 1 |

              2        3
o6 : Matrix ZZ  <--- ZZ
i7 : X1 = normalToricVariety V1;
i8 : set rays X1 === set rays PP2

o8 = true
i9 : max X1 === max PP2

o9 = true
i10 : isAmple (2*PP2_0)

o10 = true
i11 : V2 = vertices (2*PP2_0)

o11 = | 0 2 0 |
      | 0 0 2 |

               2        3
o11 : Matrix ZZ  <--- ZZ
i12 : X2 = normalToricVariety V2;
i13 : rays X2 === rays X1

o13 = true
i14 : max X2 === max X1

o14 = true
On a Hirzebruch surface, the polytopes associated to non-ample Cartier divisors give rise to other normal toric varieties.
i15 : FF2 = hirzebruchSurface 2;
i16 : isAmple FF2_2

o16 = false
i17 : V3 = vertices FF2_2

o17 = | 0 1 |
      | 0 0 |

               2        2
o17 : Matrix ZZ  <--- ZZ
i18 : normalToricVariety V3  -- a degenerated version of the projective line

o18 = normalToricVariety ( {{1, 0}, {-1, 0}} , {{0}, {1}} )

o18 : NormalToricVariety
i19 : isDegenerate normalToricVariety V3  

o19 = true
i20 : isAmple FF2_3

o20 = false
i21 : V4 = vertices FF2_3

o21 = | 0 0 2 |
      | 0 1 1 |

               2        3
o21 : Matrix ZZ  <--- ZZ
i22 : normalToricVariety V4 -- a weighted projective space

o22 = normalToricVariety ( {{1, 0}, {-1, 2}, {0, -1}} , {{0, 1}, {0, 2}, {1, 2}} )

o22 : NormalToricVariety
i23 : vertices FF2_1

o23 = 0

               2        1
o23 : Matrix ZZ  <--- ZZ
i24 : isAmple (FF2_2+FF2_3)

o24 = true
i25 : V5 = vertices (FF2_2+FF2_3)

o25 = | 0 1 0 3 |
      | 0 0 1 1 |

               2        4
o25 : Matrix ZZ  <--- ZZ
i26 : normalToricVariety V5 -- isomorphic Hirzebruch surface

o26 = normalToricVariety ( {{1, 0}, {-1, 2}, {0, 1}, {0, -1}} , {{0, 2}, {0, 3}, {1, 2}, {1, 3}} )

o26 : NormalToricVariety

See also