Function Fields of Real Varieties

2010-10-27T12:51:46Z

Let $V$ be a geometrically irreducible and reduced scheme defined over the real numbers, and let $K = K(V)$ be its function field.

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.
If $V$ does have real points, is $K$ necessarily formally real?

Thanks for the help.

Answer by Pete L. Clark for Function Fields of Real Varieties

2010-10-27T14:21:23Z

The theorem you want is due to Serge Lang, from the following paper:

The theory of real places. Ann. of Math. (2) 57, (1953). 378–391.

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.

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.

I also looked in Bochnak, Coste and Roy, 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.

Function Fields of Real Varieties - MathOverflow

Function Fields of Real Varieties

Answer by Pete L. Clark for Function Fields of Real Varieties