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ToricVectorBundles :: regCheck

regCheck -- checking the regularity condition for a toric vector bundle



"For a toric vector bundle in Kaneyama's description, the regularity condition means that for every pair of maximal cones $\sigma_1,\sigma_2$intersecting in a common codimension-one face, the two sets of degrees $d_1,d_2$ and the transition matrix $A_{1,2}$ fulfil the regularity condition. I.e. for every $i$ and $j$ we have that either the $(i,j)$ entry of the matrix $A_{1,2}$ is $0$ or the difference of the $i$-th degree vector of $d_1$ of $\sigma_1$ and the $j$-th degree vector of $d_2$ of $\sigma_2$ is in the dual cone of the intersection of $\sigma_1$ and $\sigma_2$."

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 automatically.

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

See also

Ways to use regCheck :

For the programmer

The object regCheck is a method function.