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PositivityToricBundles :: restrictToInvCurves

restrictToInvCurves -- computes the restrictions of a toric vector bundle to the torus invariant curves

Synopsis

Description

Given a toric vector bundle in Klyachko's description, restrictToInvCurves computes its restrictions to the torus invariant curves, which are isomorphich to a direct sum of line bundles $\mathbb P^1$. Recall that on $\mathbb P^1$ any vector bundle splits into $\oplus_i O_{\mathbb P^1}(a_i)$. Therefore restrictToInvCurves returns a hash table with keys the invariant curves and as values lists with the integers $a_i$.
By [HMP, Theorem 2.1], if all these integers$a_i$ are non-negative or positive, the original toric vector bundle is nef or ample. Hence, the methods isNef and isAmple check exactly that.
restrictToInvCurves calls internally the method toricChernCharacter; whereas isNef and isAmple are simple checks on the output of restrictToInvCurves.
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 : restrictToInvCurves E

o2 = HashTable{| -1 | => {2, 1}}
               | -1 |
               | 0 | => {2, 1}
               | 1 |
               | 1 | => {1, 2}
               | 0 |

o2 : HashTable
i3 : isNef E

o3 = true
i4 : isAmple E

o4 = true
In this example we see that the vector bundle is ample, as all integers are positive.

See also

Ways to use restrictToInvCurves :

For the programmer

The object restrictToInvCurves is a method function with options.