Generation of k-jets on Toric Varieties

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In this notes we study k-jet ample line bundles L on a non singular toric variety X, i.e. line bundles with global sections having arbitrarily prescribed k-jets at a finite number of points. We introduce the notion of an associated k-convex -support function, ψₗ, requiring that the polyhedra Pₗ has edges of length at least k. This translates to the property that the intersection of L with the invariant curves, associated to every edge, is ≥ k. We also state an equivalent criterion in terms of a bound of the Seshadri constant (L,x). More precisely we prove the equivalence of the following: (1) L is k-jet ample; (2) L· C≥ k, for any invariant curve C; (3) ψₗ is k-convex; (4) the Seshadri constant (L,x)≥ k for each x∈ X.