Generation of $k$-jets on Toric Varieties

Details Generation of $k$-jets on Toric Varieties Generation of $k$-jets on Toric Varieties

1997-10-15

arxiv.org/abs/alg-geom/9710018v1

Mathematics

Algebraic Geometry

14M25, 14J60, 14C20(14C25, 14E25), 17 pages, AmsLatex, see home page http://www.math.kth.se/~sandra/Welcome

Scientific paper

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, $\psi_L$, 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) $\psi_L$ is $k$-convex; (4) the Seshadri constant $\e(L,x)\geq k$ for each $x\in X$.

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