# Hopf lemma for line bundles on curves in algebraic geometry

In the paper http://arxiv.org/pdf/math/0110256v1.pdf Claire Voisin proves that all linear subspaces which lie inside of a (not too big) secant variety of a smooth projective curve must lie inside one of the secant planes. On page 4 of this paper the Hopf lemma is used. I would appreciate to know what exactly the Hopf lemma is, or where I can find a reference for it. As far as I understand from the context of the above paper, the Hopf lemma is a statement of the following form. Suppose $L_1$ and $L_2$ are two line bundles on a smooth projective curve $C$ and let $V$ be a subspace of $H^0(C,L_1)$. Then the rank of the multiplication map $V \otimes H^0(C,L_2)\to H^0(C, L_1 \otimes L_2)$ is at least $\dim V + \dim H^0(C,L_2) -1$, and the inequality is strict if $L_2$ is very ample.

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See the bottom of p.108 of Arbarello-Cornalba-Griffiths-Harris, Volume I. –  Yusuf Mustopa Oct 30 '13 at 1:22
Thank you very much, Yusuf! This answers my question. -OP –  user42066 Oct 30 '13 at 1:51