  Given scheme-theoretic equations for a nonsingular subvariety, we prove that
the higher cohomology groups for suitable twists of the corresponding ideal
sheaf vanish. From this result, we obtain linear bounds on the multigraded
Castelnuovo-Mumford regularity of a nonsingular subvariety, and new criteria
for the embeddings by adjoint line bundles to be projectively normal. A special
case of our work recovers the vanishing theorem of Bertram, Ein, and
Lazarsfeld.
