Description
Given a toric vector bundle in Klyachko's description,
separatesJets determines up to which order the vector bundle separates jets. Note that a toric vector bundle is globally generated or very ample, if it separates 0jets or 1jets, respectively, see [RJS, Theorem 1.2, 6.2 and 6.5]. Hence, the methods
isGloballyGenerated and
isVeryAmple only ask whether
separatesJets returns a nonnegative or positive integer, respectively.
If the vector bundle is not even globally generated, then
separatesJets returns the value
infinity.
separatesJets calls internally the methods
parliament and
toricChernCharacter; whereas
isGloballyGenerated and
isVeryAmple are simple checks on the output of
separatesJets.
i1 : E = tangentBundle(projectiveSpaceFan 2)
o1 = {dimension of the variety => 2 }
number of affine charts => 3
number of rays => 3
rank of the vector bundle => 2
o1 : ToricVectorBundleKlyachko

i2 : separatesJets E
o2 = 1
o2 : QQ

i3 : isGloballyGenerated E
o3 = true

i4 : isVeryAmple E
o4 = true

In this example, the vector bundle $\mathcal E$ separates 1jets, hence is very ample.