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# deltaE -- the polytope of possible degrees that give non zero cohomology

- Usage:
    `P = deltaE E`
- Inputs:
  - E, an instance of the type ToricVectorBundle
- Outputs:
  - P, a convex polyhedron

## Description

For a toric vector bundle over a complete toric variety there is a finite set of degrees $u$ such that the degree $u$ part of the cohomology of the vector bundle is non-zero. This function computes a polytope $\Delta_E$, such that these degrees are contained in this polytope. If the underlying toric variety is not complete then an error is returned.

    i1 : E = toricVectorBundle(2,pp1ProductFan 2, "Type" => "Kaneyama")

    o1 = {dimension of the variety => 2 }
          number of affine charts => 4
          rank of the vector bundle => 2

    o1 : ToricVectorBundleKaneyama

    i2 : P = deltaE E

    o2 = {ambient dimension => 2           }
          dimension of lineality space => 0
          dimension of polyhedron => 0
          number of facets => 0
          number of rays => 0
          number of vertices => 1

    o2 : Polyhedron

    i3 : vertices P

    o3 = 0

                  2       1
    o3 : Matrix QQ  <-- QQ

    i4 : E1 = tangentBundle projectiveSpaceFan 2

    o4 = {dimension of the variety => 2 }
          number of affine charts => 3
          number of rays => 3
          rank of the vector bundle => 2

    o4 : ToricVectorBundleKlyachko

    i5 : P1 = deltaE E1

    o5 = {ambient dimension => 2           }
          dimension of lineality space => 0
          dimension of polyhedron => 2
          number of facets => 6
          number of rays => 0
          number of vertices => 6

    o5 : Polyhedron

    i6 : vertices P1

    o6 = | -1 1 0  1  0 -1 |
         | 0  0 -1 -1 1 1  |

                  2       6
    o6 : Matrix QQ  <-- QQ

## See also

- eulerChi -- the Euler characteristic of a toric vector bundle
- HH^ZZ ToricVectorBundle -- the i-th cohomology group of a toric vector bundle
- hh^ZZ ToricVectorBundle

## Ways to use deltaE:

- `deltaE(ToricVectorBundle)`

## For the programmer

The object deltaE is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldToricVectorBundles.m2:2699:0.