Let $X \subset P^n$ 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?

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?