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}}) |
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 |
i6 : D1 o6 = 2*X1 + 2*X1 1 3 o6 : ToricDivisor on X1 |
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 |
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}}) |
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 |
i13 : D2 o13 = 3*X2 + 2*X2 1 3 o13 : ToricDivisor on X2 |
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 |
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}}) |
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 |
i20 : D3 o20 = 3*X3 + X3 + X3 + 3*X3 + 2*X3 + 4*X3 0 1 4 5 6 7 o20 : ToricDivisor on X3 |
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 |
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}}) |
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 |
i27 : D4 o27 = X4 + 2*X4 1 3 o27 : ToricDivisor on X4 |
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 |
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}}) |
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 |
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 |
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}}) |
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 |
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 |
We thank Milena Hering for her help creating the database.