Let $k$ be a field. $N,d$ be positive integers and define $L= {N+d \choose d } (N+1)-1 $ .
Then can we identify $\mathbb P^L$ with the space of $N+1$- tuples of homogeneous polynomials of degree $d$ in $N+1$ variables , with at least one polynomial non-zero, such that each $f=[f_0,...,f_N] \in \mathbb P^L$ defines a rational map $ f : \mathbb P^N \to \mathbb P^N$ ?
Note: I am trying to understand the first few lines of the introduction of this paper
Homogeneous polynomials of degree $d$ in $N+1$ variables form a vector space $V$ of dimension $\binom{N+d}{d}$ by stars and bars. $(N+1)$-tuples of these form a vector space $V^{N+1}$ of dimension $\binom{N+d}{d}(N+1)$, and any two $(N+1)$-tuples of polynomials which are all the same scalar multiple of eachother determine the same rational morphism $\Bbb P^N\dashrightarrow \Bbb P^N$ by sending $[x_0:\cdots:x_N]\mapsto [f_0(x_0,\cdots,x_n):\cdots:f_N(x_0,\cdots,x_N)]$. So it makes sense to consider the projective space of such polynomials, which is isomorphic to $\Bbb P^L$ (as the projective space on a $E$-dimensional vector space is isomorphic to $\Bbb P^{E-1}$) after choosing a basis of the vector space $V^{N+1}$.
The morphisms that these equivalence classes of polynomials determine are rational because it cannot be guaranteed that there are no points at which all the polynomials vanish - for instance, the $(N+1)$-tuple $(x,0,0,\cdots,0)$ vanishes on an entire $\Bbb P^{N-1}$.
This technique (and it's generalizations) of using the projectivization of the vector space of polynomials of a certain degree to give geometric content to the space of maps between two projective spaces is something that shows up a fair amount in algebraic geometry - I'm kind of shocked that there was no duplicate around for this question.