Local positivity of line bundles on smooth toric varieties and Cayley polytopes

For any positive integer k the k-th osculating dimension at a given point x of a variety X embedded in projective space gives a measure of the local positivity of order k at that point. In this paper we show that a smooth toric embedding having the property that at every point the t  -th osculating dimension is maximal if and only if t≤kt≤k, is associated to a Cayley polytope of order k. This result generalises an earlier characterisation by David Perkinson. In addition we prove that the above assumptions are equivalent to requiring that the Seshadri constant is exactly k at every point of X, generalising a result of Atsushi Ito.