Scheme-theoretic account of why every variety embeds in a complete variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T13:22:26Z http://mathoverflow.net/feeds/question/56947 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56947/scheme-theoretic-account-of-why-every-variety-embeds-in-a-complete-variety Scheme-theoretic account of why every variety embeds in a complete variety Charles Staats 2011-03-01T00:36:26Z 2011-03-01T11:04:38Z <p>The standard reference for the statement that "any abstract variety is an open subscheme of a complete variety" is Nagata's 1962 paper <i>Imbedding of an abstract variety in a complete variety</i>. 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 scheme-theoretic language (or other language more comprehensible to the contemporary reader)?</p> http://mathoverflow.net/questions/56947/scheme-theoretic-account-of-why-every-variety-embeds-in-a-complete-variety/56949#56949 Answer by Keenan Kidwell for Scheme-theoretic account of why every variety embeds in a complete variety Keenan Kidwell 2011-03-01T00:46:16Z 2011-03-01T00:46:16Z <p>Brian Conrad has a writeup on this:</p> <p><a href="http://math.stanford.edu/~conrad/papers/nagatafinal.pdf" rel="nofollow">http://math.stanford.edu/~conrad/papers/nagatafinal.pdf</a></p> http://mathoverflow.net/questions/56947/scheme-theoretic-account-of-why-every-variety-embeds-in-a-complete-variety/56988#56988 Answer by Leo Alonso for Scheme-theoretic account of why every variety embeds in a complete variety Leo Alonso 2011-03-01T11:04:38Z 2011-03-01T11:04:38Z <p>Apart from Brian's, published as:</p> <p>Deligne's notes on Nagata compactifications. J. Ramanujan Math. Soc. 22 (2007), no. 3, 205–257.</p> <p>there are:</p> <p>Lütkebohmert, On compactification of schemes. Manuscripta Math. 80 (1993), no. 1, 95–111.</p> <p>and</p> <p>Vojta: Nagata's embedding theorem, arXiv:0706.1907</p> <p>and, finally</p> <p>Deligne: Le théorème de plongement de Nagata, Kyoto J. Math. 50, Number 4 (2010), 661-670.</p> <p>All of them are worth reading. The issue is certainly subtle and important, at least for cohomological constructions.</p>