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

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?

• Without further data, "degree" only makes sense for line bundles (=invertible sheaves) on curves. Commented Jun 21, 2015 at 21:39
• I have edited the question. Use the definition of degree as in Ex. II. 6.2 of Hartshorne after identifying invertible sheaves with corresponding divisors as explained in chapter II.6 of Hartshorne. Commented Jun 21, 2015 at 22:51
• @user54369 : But Hartshorne Ex. II. 6.2 , isnt written for curves? Commented Jun 22, 2015 at 6:15
• @user54369 : it is Ex II. 6.12 the one for curves, sorry. But if you use Hartshorne definition of degree (i.e. Ex II. 6.2.) it seems to me that your claim follows from the 1-1 correspondence of Proposition 13 plus the fact that the zero set of a section is an effective divisor whose degree seems to be non negative. I think that the last follows also from Ex. II. 6.2. part c). Commented Jun 22, 2015 at 6:32
• Let me try to expand on my comments.That exercise in Hartshorne is talking about the degree of a divisor on a variety with a fixed embedding in projective space. That is the "extra data" I was referring to. This sounds like a petty point to argue over, but in fact it really matters: there are varieties $X$ which come with many different embeddings into projective space, and depending on which embedding you choose, you can get a different answer for the degree of a subvariety. In other words, this notion of degree is not intrinsic to $X$ (unlike the case for curves). Commented Jun 22, 2015 at 22:51

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.

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

• But isnt this written for curves? Commented Jun 21, 2015 at 20:16
• Yes, sorry. I am going to edit my answer. Commented Jun 21, 2015 at 20:52
• I am assuming that the variety is projective as well Commented Jun 21, 2015 at 23:00