Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 scheme-theoretic language (or other language more comprehensible to the contemporary reader)?

share|cite|improve this question
Brian Conrad has a modern version, which you can get off his website. I think there are others as well. –  Donu Arapura Mar 1 '11 at 0:46

2 Answers 2

up vote 9 down vote accepted

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.


Vojta: Nagata's embedding theorem, arXiv:0706.1907

and, finally

Deligne: Le théorème de plongement de Nagata, Kyoto J. Math. 50, Number 4 (2010), 661-670.

All of them are worth reading. The issue is certainly subtle and important, at least for cohomological constructions.

share|cite|improve this answer

Brian Conrad has a writeup on this:

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.