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

## Synopsis

• Usage:
l = restrictToInvCurves E
b = isNef E
b = isAmple E
• Inputs:
• Outputs:
• l,
• b,
• Consequences:
• The result of restrictToInvCurves will be stored as a cacheValue in E.cache#restrictionsToInvCurves. It will be used by other methods.

## 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.

## Ways to use restrictToInvCurves :

• "restrictToInvCurves(ToricVectorBundleKlyachko)"

## For the programmer

The object restrictToInvCurves is .