  Using multigraded Castelnuovo-Mumford regularity, we study the equations
defining a projective embedding of a variety X. Given globally generated line
bundles B_1, ..., B_k on X and integers m_1, ..., m_k, consider the line bundle
L := B_1^m_1 \otimes ... \otimes B_k^m_k. We give conditions on the m_i which
guarantee that the ideal of X in P(H^0(X,L)) is generated by quadrics and the
first p syzygies are linear. This yields new results on the syzygies of toric
varieties and the normality of polytopes.
