I have the impression that exercises started to appear on research level books not too long ago, say the 60s or even the 70s...

I guess you need at least that your intersection of quadrics contains a line (which I think is not always the case over the reals), otherwise I wouldn't know how to do that. In that case the map from $\mathbb{P}^3$ to $X$ your intersection should go like this. Take a $\mathbb{P}^3$ disjoint from the line $L$ and pick $p\in \mathbb{P}^3$. Then consider the $\mathbb{P}^2=:Q$ spanned by $L$ and p. If all goes well, each quadric intersects $Q$ in two degenerate conics $L+M$ and $L+N$ with $M$ and $N$ lines. The coordinates of $M\cap N$ are the image of $p$. This is just a rational map, btw.

@abx: yeah, that's what I thought as well. unfortunately, browsinf the paper I have found no trace of such change of convention. Maybe they simply tacitely assume opposite conventions?

Yes, thanks, I looked at that and it is very well explained. Actually the invariants are much simpler than I expected.

@Jason, Yes sorry the lines are ordered as usual. But by naive dimension count the quotient should have dimension 1, am I wrong?