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PositivityToricBundles :: groundSet

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

Synopsis

Description

Given a toric vector bundle E in Klyachko’s description on a toric variety X = TV(Σ), it is encoded by increasing filtrations Eρ(j) for each ray ρ∈Σ(1). To these filtrations we can associated the set L(E) of intersections ρ Eρ (jρ), where (jρ)ρ runs over all tuples in Σ(1). This set L(E) is ordered by inclusion and there is a unique matriod M(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.

See also

Ways to use groundSet :