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NormalToricVarieties :: isVeryAmple(ToricDivisor)

isVeryAmple(ToricDivisor) -- whether a torus-invariant Weil divisor is very ample

Synopsis

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. On a normal toric variety, the following are equivalent:
  • D is a very ample divisor;
  • for the associated lattice polytope P and every vertex mi ∈P, the semigroup ℕ(P ∩M - mi) is saturated in the group characters M.

On a smooth normal toric variety every ample divisor is very ample.

i1 : PP3 = projectiveSpace 3;
i2 : isAmple PP3_0

o2 = true
i3 : isVeryAmple PP3_0

o3 = true
i4 : FF2 = hirzebruchSurface 2;
i5 : isAmple (FF2_2+FF2_3)

o5 = true
i6 : isVeryAmple (FF2_2+FF2_3)

o6 = true
A Cartier divisor is ample when some positive integer multiple is very ample. On a normal toric variety of dimension d the (d-1) multiple of any ample divisor is always very ample.
i7 : X = normalToricVariety matrix {{0,1,0,0,1},{0,0,1,0,1},{0,0,0,1,1},{0,0,0,0,3}};
i8 : dim X

o8 = 4
i9 : D = 3*X_0

o9 = 3*D
        0

o9 : ToricDivisor on X
i10 : isAmple D

o10 = true
i11 : isVeryAmple D

o11 = false
i12 : isVeryAmple (2*D)

o12 = false
i13 : isVeryAmple (3*D)    

o13 = true

See also