Let $k_1, k_2, k_0 \in \mathbb{N}$.
If $k_1 = k_2 = k_0$ then the map
which sends $[x_0:x_1:x_2] \in \mathbb{P}^2$ to $[x_0^{k_0}:x_1^{k_1}:x_2^{k_2}] \in \mathbb{P}^2$ is well defined, but
it's not well defined if all of the $k_i$'s are not the same.
(My understnading of morphisms between projective varieties:
$\varphi: V \to W$ is a map between projective varieties $V \subset \mathbb{P}^n$ and $W \subset \mathbb{P}^m$ given by $\varphi([x_0 : \ldots : x_n]) = [\varphi_0([x_0 : \ldots : x_n]): \ldots : \varphi_m([x_0 : \ldots : x_n])]$, where the $\varphi_i$ are homogeneous polynomials of the same degree that don't vanish simultaneously at any point of $V$.)
I was just wondering when not all of the $k_i$'s are equal, does the map 'make sense' if I change the domain to affine space?
i.e. if I define a map from $\mathbb{A}^3 \backslash \{ \mathbf{0} \}$ to $\mathbb{P}^2$ by $(x_0, x_1, x_2) \rightarrow [x_0^{k_0}:x_1^{k_1}:x_2^{k_2} ]$, is this just a weird map that sends of $\mathbb{A}^3 \backslash \{ \mathbf{0} \}$ to $\mathbb{P}^2$? or does this become a morphism in an appropriate category (something that generalizes affine varieties and projective varities maybe?) Any comments would be appreciated. Thank you.
ps for simplicity I'm only thinking the affine space and the projective space over $\mathbb{C}$
Best Answer
You can consider the affine map $(x,y,z)\mapsto (x^a,y^b,z^c)$ from $\mathbb{A}^3-\{0\}$ to itself. (Observe that this map does not contain $0$ in its image). Hence you can compose it with the quotient map $(x,y,z)\mapsto [x:y:z]$ (which is well-defined on non-zero points) to obtain the map you give.