I have a question about a conclusion in the proof of lemma 7.4.3 from Liu's "Algebraic Geometry and Arithmetic Curves" (on page 286):
We assume that the invertible sheaf $O_{X}(D)$ is globally generated by $s_0,...,s_n \in L(D) =H^0(X,O_{X}(D))$. This fact gives rise for a morphism $f:X \to \mathbb{P}^n$ induced by the $s_i$.
Let $s \in H^0(X, O_{X}(D))/H^0(X, O_{X}(D))$ a section with $s = \sum_{0 \le i \le n}\lambda_i s_i$.
Let $f(p)=(p_0,..., p_n)$.
Since $O_{X}(D)$ locally free the stalk at $p$ is isomorphic to $O_{X,p}$ via $O_{X,p} \to O_{X}(D)_p, r \mapsto r \cdot e$ (where $e$ basis of $O_{X}(D)_p$).
We observe $(s_i)_p \in p_i \cdot e +\mathfrak{m}^2_p$.
The QUESTION is why does this imply $\sum_i \lambda_i p_i =0$?
I don't understand it.