I was wondering why we hardly ever talk about the rational projective spaces. We frequently use the real, complex, finite or even quaternionic ones, but for instance there is apparently only one question on M.SE about $\mathbb{Q}P^{n}$.
Is it true that $\mathbb{Q}P^{n}$ less studied/less useful than the other projective spaces? Is there any particular reason for this?
I can try to make my thoughts more clear/more precise if necessary. But I hope I'm not the only one who have rarely seen $\mathbb{Q}P^{n}$.
Thank you for your comments!
Algebraic geometers in fact study an object that is just called $\mathbb{P}^n$. It is a scheme, and it contains the data of $K \mathbb{P}^n$ for any field $K$ (but is more general than this, e.g. $K$ can be replaced by an arbitrary commutative ring; at this level of generality the definition becomes more subtle).
As mentioned in the comments, what makes $\mathbb{RP}^n$ and $\mathbb{CP}^n$ (and $\mathbb{HP}^n$ and, when $n = 1, 2$, $\mathbb{OP}^n$) special is that they are also smooth manifolds, so are also studied by algebraic topologists, differential geometers, etc.
$\mathbb{QP}^1$ in particular also makes an appearance in the theory of modular forms as the set of cusps.