Nagata’s Embedding Theorem

In 1962–63, M. Nagata showed that an abstract variety could be embedded into a complete variety. Later, P. Deligne translated Nagata’s proof into the language of schemes, but did not publish his notes. This paper, which is to appear as an appendix in a forthcoming book, gives an elaboration of Deligne’s notes. It also contains some complementary results on extending divisors and vector sheaves to suitable completions. The goal of this note is to prove that, if X is a scheme, separated and of finite type over a noetherian scheme S , then there exists a proper S-scheme X and an open immersion X →֒ X over S with schematically dense image (Theorem 4.1). In addition, given a Cartier divisor D or a vector sheaf E on X , the completion X can be chosen so that D or E extends to a Cartier divisor or vector sheaf (respectively) on X . The first assertion was proved by Nagata; see [N 1], [N 2]. Nagata’s proof is phrased in terms of Zariski’s language of algebraic geometry, though, which makes it difficult for many to read. Because of this, P. Deligne wrote some notes [D], which translate Nagata’s work into the language of schemes. These notes, however, are unpublished. This paper, which is based closely on Deligne’s notes, was written to give a mostly self-contained exposition of the proof. After writing these notes, I encountered another (much more thorough) rendition of Deligne’s notes by B. Conrad [C]. Conrad also notes the existence of another proof of Nagata’s theorem by Lütkebohmert [L], which uses schemes but uses different methods. Sections 1–4 of this note give Nagata’s proof, following [D]. Section 5 adds some complementary results on constructing the completion so that certain sheaves or divisors extend to the completion. Throughout this note all schemes are assumed to be noetherian and all ideal sheaves to be coherent. Projective morphisms are as defined in [EGA]. Schemes are not assumed to be separated unless it is explicitly mentioned. 2000 Mathematics Subject Classification. Primary 14A15; Secondary 14E99, 14J60. Supported by NSF grants DMS95-32018 and DMS-0500512, and the Institute for Advanced Study. 1