  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.
