Cyclic coverings and higher order embeddings of algebraic varieties

An algebraic variety X is embedded to the order k via a line bundle L if the global sections of L generate all (simultaneous) jets of order k on X or if they separate all zero-dimensional subschemes of length at most k+1. Even though we refer to both situations as "higher order embeddings", the first notion (in which case L is said to be k-jet ample) is stronger than the second one (when L is k-very ample). The purpose of this paper is to study higher order embeddings of cyclic coverings \pi:Y\to X via line bundles given by pulling back "sufficiently positive" line bundles on X. Given a line bundle L on X, we relate the order of the embedding defined by \pi^*L to that of L and of certain rank 1 summands of the vector bundle L\tensor\pi_*\calo_Y. As expected, the sufficient conditions for \pi^*L to be k-jet ample are stronger then the ones needed in order for \pi^*L to be k-very ample.