"For a toric vector bundle in Kaneyama’s description, the regularity condition means that for every pair of maximal cones
σ1,σ2intersecting in a common codimension-one face, the two sets of degrees
d1,d2 and the transition matrix
A1,2 fulfil the regularity condition. I.e. for every
i and
j we have that either the
(i,j) entry of the matrix
A1,2 is
0 or the difference of the
i-th degree vector of
d1 of
σ1 and the
j-th degree vector of
d2 of
σ2 is in the dual cone of the intersection of
σ1 and
σ2."
Note that this is only necessary for toric vector bundles generated 'by hand' using
addBaseChange and
addDegrees, since bundles generated for example by
tangentBundle satisfy the condition autmatically.
i1 : E = tangentBundle(pp1ProductFan 2,"Type" => "Kaneyama")
o1 = {dimension of the variety => 2 }
number of affine charts => 4
rank of the vector bundle => 2
o1 : ToricVectorBundleKaneyama
|
i2 : regCheck E
o2 = true
|