The goal of this article is to study the equations and syzygies of embeddings
of rational surfaces and certain Fano varieties. Given a rational surface X and
an ample and base-point-free line bundle L on X, we give an optimal numerical
criterion for L to satisfy property Np. This criterion turns out to be a
characterization of property Np if X is anticanonical. We also prove syzygy
results for adjunction bundles and a Reider type theorem for higher syzygies.
For certain Fano varieties we also prove results on very ampleness and higher
syzygies.Comment: 26 pages, AMSTe
