On dimension of rational curves in $\mathbb P^r$ Let us consider $X:=$ space of smooth rational curves of degree $d$ in $\mathbb P^r$. Then I have come across the following statement " $X$ is a smooth quasi-projective variety of dimension $(r+1)(d+1)-4$."
Can someone give me a reference where this fact is described or a short proof of the same.
 A: It's by counting polynomials.
A smooth rational curve $C \subseteq \mathbb{P}^r$ of degree $d$ can be described, non-uniquely, as the image of a map $\mathbb{P}^1 \to \mathbb{P}^r$ of the form $$[s:t] \in \mathbb{P}^1 \mapsto [f_0(s,t) : \cdots : f_r(s,t)]$$ given by $r+1$ homogeneous, degree-$d$ polynomials in $s,t$, with no common factor.
The space of degree-$d$ homogeneous polynomials in two variables is $(d+1)$-dimensional and we're choosing $r+1$ of them. This is overcounting, but so far it gives a $(d+1)(r+1)$-dimensional space (literally $\mathbb{A}^{(d+1)(r+1)}$) of choices.
The condition "no common factor" corresponds to a dense open subset $U$ of this space, so restricting to that subset doesn't change the dimensionality.
More importantly though:

*

*The choice of polynomials itself is overcounting. We could rescale the polynomials by a common factor and this would give the same map $\mathbb{P}^1 \to \mathbb{P}^r$, so we should subtract one (replace our open subset $U \subseteq \mathbb{A}^{(r+1)(d+1)}$ by the corresponding subset $U'$ of the projectivization $\mathbb{P}^{(r+1)(d+1)-1}$). Now we're no longer over-describing the map.


*But also, our goal was to count curves $C \subseteq \mathbb{P}^r$, and the map $\mathbb{P}^1 \to \mathbb{P}^r$ could be altered by an automorphism on the $\mathbb{P}^1$ side, without changing the image curve in $\mathbb{P}^r$. So we should subtract the dimension of $\mathrm{Aut}(\mathbb{P}^1) = PGL_2$, which is $3$. (This is replacing our subset $U' \subseteq \mathbb{P}^{(r+1)(d+1)}$ by a quotient space $U' //\ \mathrm{PGL}_2$ whose formal definition is a little subtle, using geometric invariant theory.)
Summarizing, we've reduced our dimension to $(r+1)(d+1)-4$.
