Generation of $k$-jets on Toric Varieties

Abstract

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 $\D$-support function, $ψ_L$, requiring that the polyhedra $P_L$ 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 $\geq k$. We also state an equivalent criterion in terms of a bound of the Seshadri constant $\e(L,x)$. More precisely we prove the equivalence of the following: (1) $L$ is $k$-jet ample; (2) $L\cdot C\geq k$, for any invariant curve $C$; (3) $ψ_L$ is $k$-convex; (4) the Seshadri constant $\e(L,x)\geq k$ for each $x\in X$.