Type: Article
Publication Date: 2001-01-19
Citations: 32
DOI: https://doi.org/10.1515/crll.2001.068
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 N p .This criterion turns out to be a characterization of property N p 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. IntroductionThe goal of this article is to study the equations and the syzygies of embeddings of rational surfaces and certain Fano varieties.Previously Butler, Homma, Kempf, and the authors had proved results regarding syzygies of (geometrically) ruled surfaces and surfaces of nonnegative Kodaira dimension.We will be interested in knowing under what conditions the resolution of the homogeneous coordinate ring SaI of an embedded variety is ``simple''.More precisely we want to know under what conditions the so-called property N p after M. Green is satis®ed.We de®ne this property next: De®nition 0.1.Let X be a projective variety.A very ample line bundle L is said to satisfy property N 0 if jLj embeds X as a projectively normal variety.A very ample line bundle L satis®es property N 1 if L satis®es property N 0 and the homogeneous ideal I of the image of X embedded by jLj is generated by quadratic equations.Finally a very ample line bundle L is