I'm trying to reconcile two definitions I've seen of a rational map between varieties. Here's the usual one from Hartshorne:
Let $X$, $Y$ be varieties. Then a rational map $\varphi : X \rightarrow Y$ is an equivalence class of pairs $\langle U, \varphi_U \rangle$ where $U$ is a non-empty open subset of $X$, $\varphi_U$ is a morphism from $U$ to $Y$, and where $\langle U, \varphi_U \rangle$, $\langle V, \varphi_V \rangle$ are equivalent if $\varphi_U$, $\varphi_V$ agree on $U \cap V$.
Versus Silverman:
Let $V_1, V_2 \subset \mathbb P^n$ be projective varieties. A rational map from $V_1$ to $V_2$ is a map of the form $\phi : V_1 \rightarrow V_2$, $\phi = [f_0, ..., f_n]$, where the functions $f_i \in \overline K(V_1)$ have the property that for every $P \in V_1$ at which $f_0(P), ..., f_n(P)$ are all defined $\phi(P) \in V_2$.
My first concern is that Silverman's definition doesn't seem to rule out $\phi := [0,...,0]$, which is not a morphism on any non-empty open subset. So I think I need to add something to Silverman's definition to make these equivalent. Can anyone tell me what's missing, and then how I could prove these are the same?