MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In the paper 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.

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.