Negative degree invertible sheaves on non-singular varieties have no global sections

Question

Let $X$ be a non-singular projective complex variety and $\mathcal{L}$ be an invertible sheaf on $X$ with negative degree. Is it true that $\mathcal{L}$ has no global sections? If so, can someone suggest a reference?

Answer 1

That seems to be correct, at least if you have in mind compact manifolds. See the introduction of http://www.google.it/url?sa=t&rct=j&q=&esrc=s&source=web&cd=62&ved=0CCoQFjABODw&url=ftp%3A%2F%2Fftp.math.ethz.ch%2Fhg%2FEMIS%2Fjournals%2FNYJM%2Fjdg%2Farchive%2Fvol.50%2F1_4.ps.gz&ei=DiOHVae6LKOcygOuooGQAQ&usg=AFQjCNEBF-29nDprqrWQeOXRVDa2wZ6RQg&sig2=A87X-NRzKdVC4BzeUEf2yQ

Answer 2

I guess it's a bit late but it might be useful to someone.

As emphasized by @Relapsarian, if we work with a non singular curve $C$ (I will assume moreover that it's a projective curve), then we can talk about the degree of an invertible sheaf.

The correct formulation of the O.P. question should be then:

([Proposition) An invertible sheaf of negative degree has no non-zeros global sections.

Suppose we have a negative-degree invertible sheaf $\mathcal L$ on $C$ with a non-zero section $s$. Then $s$ has no poles and probably some zeros, so $\operatorname{deg} \mathcal L\ge 0$ which contradicts the assumption.