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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.

PP3 = projectiveSpace 3;
isAmple PP3_0
isVeryAmple PP3_0
FF2 = hirzebruchSurface 2;
isAmple (FF2_2+FF2_3)
isVeryAmple (FF2_2+FF2_3)
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.
X = normalToricVariety matrix {{0,1,0,0,1},{0,0,1,0,1},{0,0,0,1,1},{0,0,0,0,3}};
dim X
D = 3*X_0
isAmple D
isVeryAmple D
isVeryAmple (2*D)
isVeryAmple (3*D)

See also