Representation of a morphism between projective varieties It is known that a morphism $f: X \rightarrow Y$ between two affine closed varieties $X \subset A^n_k,Y \subset A^m_k$ has the following form 
$$
f(x_1,\dots,x_n) = (f_1(x_1,\dots,x_n),\dots,f_m(x_1,\dots,x_n)) \in Y,
$$
where $f_1,\dots,f_m \in k[x_1,\dots,x_n].$
I am wondering if there exists a similar representation of a morphism between projective varieties if we would require $f_i$'s to be homogeneous of the same degree.
 A: Let $f:X \to Y$ be a map of projective varieties (projective, integral, separated schemes over a field $k$). Let $Y \subseteq \mathbb{P}^m$. Then $f$ extends to a map $g:X \to \mathbb{P}^m$, which is described by a line-bundle $\mathcal{L}$ on $X$ with $m+1$ sections $s_0,\ldots,s_m \in \mathcal{L}(X)$ for which
$$\phi:\mathcal{O}_X^{m+1} \to \mathcal{L}$$
given by $(a_0,\ldots,a_m) \mapsto a_0 s_0 + \cdots a_m s_m$ is surjective, that is the global sections $(s_i)$ generate $\mathcal{L}$.
Now because $X$ is an integral scheme there is an embedding $\mathcal{L} \subseteq \mathcal{K}_X$ where $\mathcal{K}_X$ is the constant sheaf of rational functions on $X$. So we can assume that $s_0,\ldots,s_m$ are rational functions $h_0,\ldots,h_m \in \mathcal{K}_X(X)$. So in a certain sense every map $f:X \to Y$ can be reduced to a family $(h_i)$ of rational functions on $X$ which have to fulfill two conditions:
1) The induced map of the $(h_i)$ that we call $g:X \to \mathbb{P}^m$ must factor through $Y$.
2) The map $g$ must be defined at every point of $X$, which gives a local condition on every $P \in X$ for the $(h_i)$.
A short summary of the explanation above and a discussion of the two conditions can be found in Silverman, The Arithmetic of Elliptic Curves I.§3 Maps between Varieties.
