I am quite confused the definition given by Shafaverich:
Definition: A regular map $f:X \rightarrow \mathbb P^n$ of an irreducible quasiproj. variety $X$ to projective space $\Bbb P^n$ is given by an $(m+1)$ truple of form $$(F_0 : \cdots : F_m)$$ of same degree in hom. coord. of $x \in \Bbb P^n$. We require that for every $x \in X$ there exists such an expression for $f$ such that $F_i(x)\not=0$ for at least one $i$.
The definition of regular maps between quasiprojective varieties allows us to define isomorphism.
Definition: We say a quasiproj. variety is isomoprhic to a closed subspace of an affine space , an affine variety
In this case, to apply the first definition, we must regard a closed subspace of an affine space as one in $\Bbb P^n$? What are we doing here?
I suppose we for "embed" $\Bbb A^n$ into $\Bbb P^n$, by one of the choice $\Bbb A_i^n$ (where $i$th coordinate is nonzero), and show it is independent? Also, are there not more ways to regard $\Bbb A^n$ as a subset in $\Bbb P^n$?