Hypersurfaces without real points - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:52:49Z http://mathoverflow.net/feeds/question/59635 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59635/hypersurfaces-without-real-points Hypersurfaces without real points jvp 2011-03-26T04:42:05Z 2011-03-26T11:48:26Z <p>Let $n, d$ be positive integers. I am interested in the open subset $\mathcal U_{n,d} \subset \mathbb P H^0 ( \mathbb P^n_{\mathbb R}, \mathcal O_{\mathbb P^n_{\mathbb R}}(d))$ corresponding to sections without real zeros. When $d$ is odd and $n$ arbitrary, the set $\mathcal U_{n,d}$ is empty since (homeogeneous) polynomials of odd degree always have (non-trivial) real roots.</p> <blockquote> <p><strong>Question.</strong> Suppose $d$ is even. How many connected components $\mathcal U_{n,d}$ has ? Do we know anything about the Betti numbers of $\mathcal U_{n,d}$ ?</p> </blockquote> <p>Indeed, motivated by this other <a href="http://mathoverflow.net/questions/58000/polynomial-contact-structures-on-rp3" rel="nofollow">question</a>, I am trying to figure out if it makes sense to ask for the number of connected components of the space of polynomial contact distributions of even degree on $\mathbb P^{2n+1}_{\mathbb R}$. </p> http://mathoverflow.net/questions/59635/hypersurfaces-without-real-points/59642#59642 Answer by Torsten Ekedahl for Hypersurfaces without real points Torsten Ekedahl 2011-03-26T08:24:07Z 2011-03-26T11:48:26Z <p>If $s$ is a non-zero section whose image lies in $\mathcal U_{n,d}$, then it has constant sign on $V^\ast:=\mathbb R^{n+1}\setminus{0}$ and after possibly multiplying by $-1$ we may assume that $s$ is strictly positive on $V^\ast$. The strictly positive $s$ form an open convex cone $C$ (we do not assume that $0$ belongs to a cone) and is hence contractible when non-empty which this one is when $d$ is even. As $C\to\mathcal U_{n,d}$ is a fibration with fibres $\mathbb R_+$ so is $\mathcal U_{n,d}$.</p>