# isAmple(ToricDivisor) -- whether a torus-invariant Weil divisor is ample

## Synopsis

• Function: isAmple
• Usage:
isAmple D
• Inputs:
• D,
• Outputs:
• , that is true if the divisor is ample

## Description

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:

• the divisor is ample;
• the real piecewise linear support function associated to the divisor is strictly convex;
• the lattice polytope corresponding to the divisor is full-dimensional and its normal fan equals the fan associated to the underlying toric variety;
• the intersection product of the divisor with every torus-invariant irreducible curve is positive.

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)