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Suppose that the homogeneous coordinates of $\mathbb{P}^{n}$ are $[x_0:\dots:x_n]$ and that $H$ is defined by the linear equation $L=\sum_i a_ix_i=0$. Let the holomorphic map $\phi$ be defined by $\phi=[f_0:\dots:f_n]$ and set $D=-\min_i \{div(f_i)\}$. if $\phi(X)$ isn't contained in the hyperplane $H$, then $\phi^{*}(H)=D+div(\sum_i a_i f_i)$.

During the proof of this lemma, the author writes: Fix a point $P \in X$ and choose $j$ such that $ord_p (f_j)=-D(p)$ is the minimum order. In this case the coordinate $x_j$ will not vanish at $P$...

Why $x_j$ doesn't vanish at $P$?

TheWanderer
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  • What is $X$, and what are the domain and codomain of $\phi$? What are $p$ and $D(p)$? – Bruno Joyal Oct 24 '13 at 15:43
  • $X$ is a compact Riemann surface, $P$ a point on $X$, $D(P)$ is the multiplicity of the divisor in $P$ and $\phi:X \mapsto \mathbb{P}^{n})$. – TheWanderer Oct 24 '13 at 15:46

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