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:
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) |