next | previous | forward | backward | up | top | index | toc | Macaulay2 website
PositivityToricBundles :: groundSet

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

Synopsis

Description

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 matriod $M(\mathcal E)$ 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 |, | 0 |, | 1 |}
      | 0 |  | 1 |  | 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 :

For the programmer

The object groundSet is a method function with options.