This method function accesses a database of equivalence classes of very ample divisors that embed their underlying smooth toric varieties into low-dimensional projective spaces.
The enumeration of the $41$ smooth projective toric surfaces embedding into at most projective $11$-space follows the classification in [Tristram Bogart, Christian Haase, Milena Hering, Benjamin Lorenz, Benjamin Nill Andreas Paffenholz, Günter Rote, Francisco Santos, and Hal Schenck, Finitely many smooth d-polytopes with n lattice points, Israel J. Math., 207 (2015) 301-329].
The enumeration of the $103$ smooth projective toric threefolds embedding into at most projective $15$-space follows [Anders Lundman, A classification of smooth convex 3-polytopes with at most 16 lattice points, J. Algebr. Comb., 37 (2013) 139-165].
The first $2$ toric divisors over a surface lie over a product of projective lines.
i1 : D1 = smallAmpleToricDivisor(2,0)
o1 = 2*D + 2*D
1 3
o1 : ToricDivisor on normalToricVariety ({{1, 0}, {-1, 0}, {0, 1}, {0, -1}}, {{0, 2}, {0, 3}, {1, 2}, {1, 3}})
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i2 : assert isVeryAmple D1 |
i3 : X1 = variety D1; |
i4 : assert (isSmooth X1 and isProjective X1) |
i5 : rays X1
o5 = {{1, 0}, {-1, 0}, {0, 1}, {0, -1}}
o5 : List
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i6 : D1
o6 = 2*X1 + 2*X1
1 3
o6 : ToricDivisor on X1
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i7 : latticePoints D1
o7 = | 0 1 2 0 1 2 0 1 2 |
| 0 0 0 1 1 1 2 2 2 |
2 9
o7 : Matrix ZZ <--- ZZ
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i8 : D2 = smallAmpleToricDivisor (2,1);
o8 : ToricDivisor on normalToricVariety ({{1, 0}, {-1, 0}, {0, 1}, {0, -1}}, {{0, 2}, {0, 3}, {1, 2}, {1, 3}})
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i9 : assert isVeryAmple D2 |
i10 : X2 = variety D2; |
i11 : assert (isSmooth X2 and isProjective X2) |
i12 : rays X2
o12 = {{1, 0}, {-1, 0}, {0, 1}, {0, -1}}
o12 : List
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i13 : D2
o13 = 3*X2 + 2*X2
1 3
o13 : ToricDivisor on X2
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i14 : latticePoints D2
o14 = | 0 1 2 3 0 1 2 3 0 1 2 3 |
| 0 0 0 0 1 1 1 1 2 2 2 2 |
2 12
o14 : Matrix ZZ <--- ZZ
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The $15$-th toric divisors on a surface lies over normal toric varieties with $8$ irreducible torus-invariant divisors.
i15 : D3 = smallAmpleToricDivisor (2,15);
o15 : ToricDivisor on normalToricVariety ({{1, 0}, {-1, 0}, {1, 2}, {0, 1}, {1, 1}, {0, -1}, {-1, -1}, {-1, -2}}, {{0, 4}, {0, 5}, {1, 3}, {1, 6}, {2, 3}, {2, 4}, {5, 7}, {6, 7}})
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i16 : assert isVeryAmple D3 |
i17 : X3 = variety D3; |
i18 : assert (isSmooth X3 and isProjective X3) |
i19 : rays X3
o19 = {{1, 0}, {-1, 0}, {1, 2}, {0, 1}, {1, 1}, {0, -1}, {-1, -1}, {-1, -2}}
o19 : List
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i20 : D3
o20 = 3*X3 + X3 + X3 + 3*X3 + 2*X3 + 4*X3
0 1 4 5 6 7
o20 : ToricDivisor on X3
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i21 : latticePoints D3
o21 = | 0 1 0 -2 -1 1 0 -3 -2 -1 -3 -2 |
| 0 0 1 1 1 1 2 2 2 2 3 3 |
2 12
o21 : Matrix ZZ <--- ZZ
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Last, $25$ toric divisors on a surface lies over Hirzebruch surfaces.
i22 : D4 = smallAmpleToricDivisor (2,30);
o22 : ToricDivisor on normalToricVariety ({{1, 0}, {-1, 0}, {0, 1}, {5, -1}}, {{0, 2}, {0, 3}, {1, 2}, {1, 3}})
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i23 : assert isVeryAmple D4 |
i24 : X4 = variety D4; |
i25 : assert (isSmooth X4 and isProjective X4) |
i26 : rays X4
o26 = {{1, 0}, {-1, 0}, {0, 1}, {5, -1}}
o26 : List
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i27 : D4
o27 = X4 + 2*X4
1 3
o27 : ToricDivisor on X4
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i28 : latticePoints D4
o28 = | 0 1 0 1 0 1 1 1 1 1 1 |
| 0 0 1 1 2 2 3 4 5 6 7 |
2 11
o28 : Matrix ZZ <--- ZZ
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The first $99$ toric divisors on a threefold embed a projective bundle into projective space.
i29 : D5 = smallAmpleToricDivisor(3,75);
o29 : ToricDivisor on normalToricVariety ({{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {1, 1, 0}, {0, -1, 0}, {-1, -1, 0}, {0, 0, 1}, {0, 0, -1}}, {{0, 3, 6}, {0, 3, 7}, {0, 4, 6}, {0, 4, 7}, {1, 2, 6}, {1, 2, 7}, {1, 5, 6}, {1, 5, 7}, {2, 3, 6}, {2, 3, 7}, {4, 5, 6}, {4, 5, 7}})
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i30 : assert isVeryAmple D5 |
i31 : X5 = variety D5; |
i32 : assert (isSmooth X5 and isProjective X5) |
i33 : assert (# rays X5 === 8) |
i34 : D5
o34 = 2*X5 + X5 + 2*X5 + X5 + X5
0 3 4 5 7
o34 : ToricDivisor on X5
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i35 : latticePoints D5
o35 = | 0 -1 0 -2 -1 -2 -1 0 -1 0 -2 -1 -2 -1 |
| 0 0 1 1 1 2 2 0 0 1 1 1 2 2 |
| 0 0 0 0 0 0 0 1 1 1 1 1 1 1 |
3 14
o35 : Matrix ZZ <--- ZZ
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The last $4$ toric divisors on a threefold embed a blow-up of a projective bundle at few points into projective space.
i36 : D6 = smallAmpleToricDivisor (3,102);
o36 : ToricDivisor on normalToricVariety ({{1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {1, 1, 1}, {-1, 0, -1}, {-1, -1, -1}}, {{0, 1, 4}, {0, 1, 5}, {0, 2, 3}, {0, 2, 6}, {0, 3, 4}, {0, 5, 6}, {1, 3, 4}, {1, 3, 5}, {2, 3, 6}, {3, 5, 6}})
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i37 : assert(isVeryAmple D6) |
i38 : X6 = variety D6; |
i39 : assert (isSmooth X6 and isProjective X6) |
i40 : assert (# rays X6 === 7) |
i41 : D6
o41 = X6 + 2*X6 + X6 + 2*X6
0 2 5 6
o41 : ToricDivisor on X6
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i42 : latticePoints D6
o42 = | 0 1 0 -1 1 0 -1 0 -1 0 -1 -1 -1 -1 |
| 0 0 1 1 1 2 2 0 0 1 1 2 0 1 |
| 0 0 0 0 0 0 0 1 1 1 1 1 2 2 |
3 14
o42 : Matrix ZZ <--- ZZ
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We thank Milena Hering for her help creating the database.