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
D on a complete normal toric variety, the following conditions are equivalent:
- D is ample;
- the real piecewise linear support function associated to D is strictly convex;
- the lattice polytope corresponding to D is full-dimensional and its normal fan equals the fan associated to the underlying toric variety;
- the intersection product of D with every torus-invariant irreducible curve is positive.
On projective space, every torus-invariant prime divisor is ample.
i1 : PP3 = projectiveSpace 3;
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i2 : all(#rays PP3, i -> isAmple PP3_i)
o2 = true
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On a Hirzebruch surface, none of the torus-invariant prime divisors are ample.
i3 : X1 = hirzebruchSurface 2;
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i4 : any(#rays X1, i -> isAmple X1_i)
o4 = false
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i5 : D = X1_2 + X1_3
o5 = D + D
2 3
o5 : ToricDivisor on X1
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i6 : isAmple D
o6 = true
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i7 : isProjective X1
o7 = true
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A normal toric variety is Fano if and only if its anticanonical divisors, namely minus the sum of its torus-invariant prime divisors, is ample.
i8 : X2 = smoothFanoToricVariety(3,5);
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i9 : K = toricDivisor X2
o9 = - D - D - D - D - D - D
0 1 2 3 4 5
o9 : ToricDivisor on X2
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i10 : isAmple (- K)
o10 = true
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i11 : X3 = kleinschmidt(9,{1,2,3});
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i12 : K = toricDivisor X3
o12 = - D - D - D - D - D - D - D - D - D - D - D
0 1 2 3 4 5 6 7 8 9 10
o12 : ToricDivisor on X3
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i13 : isAmple (-K)
o13 = true
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