# Rationality of intersection of quadrics

Let $X \subset \mathbb{P}^n$ be a complete intersection of two quadrics. It is classical that, if $X$ contains a line, then it is rational. The proof is very simple and basically it is given by taking the projection off the line.

It is also stated in a paper by Colliot-Thélène, Sansuc and Swinnerton-Dyer that if $X$ contains a curve of odd degree, then it is rational as well. Is there a geometric proof of this?

Are there known examples of complete intersections of two quadrics that contain a curve of odd degree and not a line?

And finally: are there other sufficient conditions for the rationality of such a variety?

-

By a theorem of Amer, an intersection of quadrics $f = g = 0$ contains a linear space of dimension $r$ over $k$ iff the quadric given by $f + tg = 0$ contains a linear space of dimension $r$ over $k(t)$. Hence it suffices to prove that any quadric which contains a curve of odd degree actually contains a line.