A Cartier divisor is very ample when it is basepoint free and the map arising from its complete linear series is a closed embedding. A Cartier divisor is ample when some positive integer multiple is very ample. For a torus-invariant Cartier divisor on a complete normal toric variety, the following conditions are equivalent:
On projective space, every torus-invariant irreducible divisor is ample.
i1 : PP3 = toricProjectiveSpace 3; |
i2 : assert all (# rays PP3, i -> isAmple PP3_i) |
On a Hirzebruch surface, none of the torus-invariant irreducible divisors are ample.
i3 : X1 = hirzebruchSurface 2; |
i4 : assert not any (# rays X1, i -> isAmple X1_i) |
i5 : D = X1_2 + X1_3 o5 = X1 + X1 2 3 o5 : ToricDivisor on X1 |
i6 : assert isAmple D |
i7 : assert isProjective X1 |
A normal toric variety is Fano if and only if its anticanonical divisors, namely minus the sum of its torus-invariant irreducible divisors, is ample.
i8 : X2 = smoothFanoToricVariety (3,5); |
i9 : K = toricDivisor X2 o9 = - X2 - X2 - X2 - X2 - X2 - X2 0 1 2 3 4 5 o9 : ToricDivisor on X2 |
i10 : assert isAmple (- K) |
i11 : X3 = kleinschmidt (9,{1,2,3}); |
i12 : K = toricDivisor X3 o12 = - X3 - X3 - X3 - X3 - X3 - X3 - X3 - X3 - X3 - X3 - X3 0 1 2 3 4 5 6 7 8 9 10 o12 : ToricDivisor on X3 |
i13 : assert isAmple (-K) |