Function Fields of Real Varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:58:17Z http://mathoverflow.net/feeds/question/43796 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43796/function-fields-of-real-varieties Function Fields of Real Varieties unknown 2010-10-27T12:51:46Z 2010-11-30T13:15:29Z <p>Let \$V\$ be a geometrically irreducible and reduced scheme defined over the real numbers, and let \$K = K(V)\$ be its function field.</p> <ol> <li><p>If \$V\$ does not have any real points, is it true that \$K\$ is not formally real? It seems this is a theorem due to (E.) Artin but I cannot find a modern reference and my German needs a little work.</p></li> <li><p>If \$V\$ does have real points, is \$K\$ necessarily formally real?</p></li> </ol> <p>Thanks for the help.</p> http://mathoverflow.net/questions/43796/function-fields-of-real-varieties/43801#43801 Answer by Pete L. Clark for Function Fields of Real Varieties Pete L. Clark 2010-10-27T14:21:23Z 2010-11-30T13:15:29Z <p>The theorem you want is due to Serge Lang, from the following paper:</p> <blockquote> <p>The theory of real places. Ann. of Math. (2) 57, (1953). 378–391. </p> </blockquote> <p>The statement is almost, but not quite, what you suggest. To see the problem, a good example to consider is the affine plane curve over \$\mathbb{R}\$ defined by \$\mathbb{R}[x,y]/(y^2+x^2+x^4)\$. This defines a geometrically integral curve over \$\mathbb{R}\$ with non-formally real fraction field but possessing an \$\mathbb{R}\$-point, namely \$(0,0)\$. The key is that \$(0,0)\$ is the only \$\mathbb{R}\$-point on this curve and (thus!) it is a singular point.</p> <p>So the correct result is that the function field of an integral affine variety \$V_{/\mathbb{R}}\$ is formally real iff \$V\$ admits a nonsingular \$\mathbb{R}\$-point. (Note that projective real algebraic varieties are also affine(!!).) Probably you could extend this to finite-type integral schemes without any trouble.</p> <p>I also looked in <em>Bochnak, Coste and Roy</em>, following Thierry Zell's suggestion, but only found "half" of this result, namely the Artin-Lang Homomorphism Theorem. It seems likely though that I just didn't look hard enough: perhaps someone can enlighten me. </p>