Let $K$ be an algebraically closed field.
Definition A rational map $$\phi = [f_0,\dots,f_n]: V_1 \to V_2$$ (where $V_1, V_2 \subseteq \mathbb{P}^n$ are projective varieties) is called regular at $P \in V_1$ if there is a function $g \in K(V_1)$ such that
- each $gf_i$ is regular at P, and
- $(gf_i)(P) \neq 0$ for some $i$.
In Silverman's book The Arithmetic of Elliptic Curves, we set $$ \phi(P) = [(gf_0)(P),\dots,(gf_n)(P)] $$ in this case.
Question Why is this definition well-defined? In particular, why does another element $h \in K(V_1)$ satisfying 1. and 2. give the same value?
Approach
- I tried to rewrite $$gf_i = \frac{p_i}{q_i}$$ and $$hf_i = \frac{r_i}{s_i}$$ where $p_i,q_i,r_i,s_i \in K[V_1]$ such that $q_i(P) \neq 0$ and $s_i(P) \neq 0$ since 1. must be satisfied. I wanted to show that if $(gf_i)(P) \neq 0$, then $h f_i (P) \neq 0$ as well because this is want we want to achieve for well-definedness. But my comptations lead to nothing.
Could you please help me explain this definition? Thank you!