TITLE: some question of hartshorne book Proposition II (7.3)
QUESTION [7 upvotes]: Let $k$ be an algebraically closed field, let $X$ be a projective scheme over $k$ and let $f:X\rightarrow \mathbb{P}^n_k$ be a morphism over $k$. Let $\mathcal{L}$ be an invertible sheaf generated global sections $s_0, \dotsc s_n $ such that $\mathcal{L} \cong f^* \mathcal{O}(1)$ and $f^*(x_i)=s_i$ for $i=0, \dotsc, n$.
Let $V \subseteq \Gamma(X, \mathcal{L})$ be the subspace spanned by the $s_i$ and suppose that for any two distinct closed points $P,Q \in X$ there is an $s\in V$ such that $s_P \in \mathfrak{m}_P\mathcal{L}_P$ but $s_Q \notin \mathfrak{m}_Q\mathcal{L}_Q$ or vice versa.
Then, in Hartshorne's book it is claimed that $f$ is clearly injective, but I don't know how to prove this fact.... Help me....

REPLY [5 votes]: Since $s\in V$, we have $s = \sum_i c_i s_i$ for some coefficients $c_i\in k$, and the condition says that $s(P) = \sum_i c_i s_i(P) = 0$ and $s(Q) = \sum_i c_i s_i(Q)\neq 0$. If it were true that $f(P) = f(Q)$, i.e., $[s_1(P):\cdots:s_n(P)] = [s_1(Q):\cdots:s_n(Q)]$, then we would have $\lambda s_i(P) = s_i(Q)$ for all $i$ and some $\lambda\in k^\times$. In particular, this gives $s(Q) = \sum_i c_i s_i(Q) = \sum_ic_i(\lambda s_i(P)) = \lambda s(P) = 0$, which contradicts the assumption on the linear series $V$.