I'm currently reading Debarre's book Higher Dimensional Algebraic Geometry. I'm having trouble understanding his argument on page 38 concerning the scheme $Mor_d(P^1_k,P^N_k)$ parametrizing morphisms of degree $d$ from the projective line over k to projective N-space. He begins by
Let $k$ be a field. Any $k$-morphism $f$ from $P^1_k$ to $P^N_k$ can be written as ...
- Isn't such a $k$-morphism merely a $k$-point in the scheme?
- This argument (which is clear to me) leads to the following conclusion (with which I have no issue):
Therefore, morphisms of degree $d$ from $P^1_k$ to $P^N_k$ are parametrized by a Zariski open set of the projective space $P((S^dk^2)^{N+1})$.
But then how does this statement on $k$-points of $Mor_d(P^1_k,P^N_k)$ lead to his following statement on the whole scheme $Mor_d(P^1_k,P^N_k)$?
We denote this quasi-projective variety by $Mor_d(P^1_k,P^N_k)$.
In other words, what happens if we begin with any scheme over $k$ instead of the base field $k$ itself? Isn't that (i.e. describing all $S$-points for all schemes $S$ over $k$) what really characterizes the scheme $Mor_d(P^1_k,P^N_k)$? Or are $k$-points of $Mor_d(P^1_k,P^N_k)$ (which are the morphisms he begins with) enough in this case? (If yes, then why do they suffice?)
Am I missing any obvious steps?