Def of rational map
Let X and Y be varieties. A rational map $\phi:X \to Y$ is an equivalence class of pairs $(U,\phi_U)$ where $U$ is a nonempty open subset of X, $\phi_U$ is a mophism of U to Y and where $(U,\phi_U)$ and $(V,\phi_V)$ are equivalent if $\phi_U$ and $\phi_V$ agree on $U \cap V$.
Question: Why a rational map is not in general a map of the set $X$ to $Y$.
Ref: Robin Hartshorne p24
I think the reason why a rational map is not a map as sets is that it dose not agree along the closed curve. image
Best Answer
First in general, we denote a rational map with a dashed arrow $\phi : X \dashrightarrow Y$.
It's an equivalence class or maps, so what does it mean : it means that there EXISTS $(U,\phi_U)$ such that your rational map is the equivalence class of this element, not that for all $U$ there is a $\phi_U$.
Imagine $U=\mathbb{C}\subseteq \mathbb{P}^1$ and $\phi_U(x)=x^2$. Then your function has a pole at infinity so it cannot be extended further : there is no $(V,\phi_V)$ with $\infty \in V$ and $\phi_U=\phi_V$ on $U \cap V$, so there is no such $(V,\phi_V)$ in the equivalence class of $(U,\phi_U)$ !