I am reading Fulton's book on algebraic curves.I am in chapter $2$ currently.After defining coordinate rings,they define polynomial maps.
If $V\subset \mathbb A^n$ and $W\subset \mathbb A^m$ are varieties then $\varphi:V\to W$ is called a polynomial map if there exist $T_1,T_2,...,T_m\in K[X_1,X_2,...,X_n]$ such that $\varphi(a_1,a_2,...,a_n)=(T_1(a_1,...,a_n),...,T_m(a_1,...,a_n))$ for all $(a_1,a_2,...,a_n)\in V$.
Now,they say that if $f$ is a polynomial function from $W\to K$ and $\varphi:V\to W$ is a polynomial map,then $f\circ \varphi$ is a polynomial function from $V\to K$.So,this gives rise to a ring homomorphism $\tilde\varphi:\Gamma(W)\to \Gamma(V)$.I wnat to understand what is the role of a polynomial map in study of algebraic curves and what things should I appreciate about these maps.I mean,what are the key points/facts about polynomial maps that I need to digest in order to proceed further?Can someone provide me some motivation as I do not have much exposure to this topic.As far as I understand,polynomial maps are the correct maps/morphisms between two affine varieties.
The proposition in section 2.2 on polynomial maps actually says that polynomial maps are precisely those that correspond to $K$-algebra homomorphisms $\Gamma(W) \to \Gamma(V)$. Much of chapter 2 is about setting up a dictionary between finitely generated reduced $K$-algebras and affine varieties, and the definition of polynomial maps (somewhat tautologically) is the analog of algebra homomorphisms.
The fact that this correspondence is also compatible with composition is the first exercise, so I suggest looking at the other exercises for some motivating properties and examples.