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groundSet -- computes the ground set of a matroid associated to a toric vector bundle



Given a toric vector bundle $\mathcal E$ in Klyachko's description on a toric variety $X = TV(\Sigma)$, it is encoded by increasing filtrations $E^{\rho}(j)$ for each ray $\rho \in \Sigma(1)$. To these filtrations we can associated the set $L(\mathcal E)$ of intersections $\cap_{\rho} E^{\rho} (j_{\rho})$, where $(j_{\rho})_\rho$ runs over all tuples in $\mathbb Z^{\Sigma(1)}$. This set $L(\mathcal E)$ is ordered by inclusion and there is a unique matroid $M(\mathcal E)$ associated to it, see [RJS, Proposition 3.1]. groundSet computes the ground set (i.e. building blocks) of this matroid.
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 : groundSet E

o2 = {| 1 |, | 1 |, | 0 |}
      | 1 |  | 0 |  | 1 |

o2 : List
With the ground set, one can compute the parliament of polytopes using parliament or compute the set of compatible bases using compatibleBases.


This method works for any toric reflexive sheaf on any toric variety.

See also

Ways to use groundSet :

For the programmer

The object groundSet is a method function with options.