The standard reference for the statement that "any abstract variety is an open subscheme of a complete variety" is Nagata's 1962 paper Imbedding of an abstract variety in a complete variety. Unfortunately, this paper was apparently written before the language of schemes became standard, and uses Nagata's own language for "algebraic geometry over a Dedekind domain." Does anyone know of a translation of this proof (or another of the same statement) into schemetheoretic language (or other language more comprehensible to the contemporary reader)?
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Apart from Brian's, published as: Deligne's notes on Nagata compactifications. J. Ramanujan Math. Soc. 22 (2007), no. 3, 205–257. there are: Lütkebohmert, On compactification of schemes. Manuscripta Math. 80 (1993), no. 1, 95–111. and Vojta: Nagata's embedding theorem, arXiv:0706.1907 and, finally Deligne: Le théorème de plongement de Nagata, Kyoto J. Math. 50, Number 4 (2010), 661670. All of them are worth reading. The issue is certainly subtle and important, at least for cohomological constructions. 


Brian Conrad has a writeup on this: 

