  We show that vanishing of asymptotic p-th syzygies implies p-very ampleness
for line bundles on arbitrary projective schemes. For smooth surfaces we prove
that the converse holds when p is small, by studying the
Bridgeland-King-Reid-Haiman correspondence for tautological bundles on the
Hilbert scheme of points. This extends previous results of Ein-Lazarsfeld,
Ein-Lazarsfeld-Yang and gives a partial answer to some of their questions. As
an application of our results, we show how to use syzygies to bound the
irrationality of a variety.
