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

## Synopsis

• Function: isVeryAmple
• Usage:
isVeryAmple D
• Inputs:
• D,
• Outputs:
• , that is true if the divisor is very 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. On a normal toric variety, the following are equivalent:

• the divisor is a very ample divisor;
• for every vertex of the associated lattice polytope associated to the divsor, the corresponding semigroup is saturated in the group characters.

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

 i1 : PP3 = toricProjectiveSpace 3; i2 : assert isAmple PP3_0 i3 : assert isVeryAmple PP3_0 i4 : FF2 = hirzebruchSurface 2; i5 : assert isAmple (FF2_2 + FF2_3) i6 : assert 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.

 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 : assert (dim X === 4) i9 : D = 3*X_0 o9 = 3*X 0 o9 : ToricDivisor on X i10 : assert isAmple D i11 : assert not isVeryAmple D i12 : assert not isVeryAmple (2*D) i13 : assert isVeryAmple (3*D)