Let $C$ be a local complete intersection projective curve in $\mathbb{P}^3$. Assume that $C$ is integral. Let $\mathcal{L}$ be a line bundle on $C$ of negative degree. We know that if $C$ is smooth then there are no global sections of $\mathcal{L}$. Is this still true if $C$ is not smooth?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
3
Yes. Of course it depends how you define the degree of $\mathcal{L}$; I recommend Mumford's Lectures on curves on an algebraic surface, Lecture 11, for a very nice approach. It implies $\deg \mathcal{O}_C(D)=\deg D :=\dim H^0(\mathcal{O}_D)$ for an effective Cartier divisor $D$. In particular, if $\mathcal{L}$ has a nonzero section $s$, then $\deg \mathcal{L}=\deg \mathrm{div}(s)\geq 0$.
-
$\begingroup$ I do not have Mumford's book avalaible now, but it seems to me that this is not quite true. If you take two intersecting lines, you can construct on it a bundle which restricts to O(1) on a line and O(-2) on the other line. This has degree -1 and h^0=1: the non trivial section will vanish on the whole second line. Indeed on reducible curves one usually consider positive bundles as bundles which have positive degree when restricted to each factor. $\endgroup$ Commented Aug 27, 2014 at 13:00
-
$\begingroup$ The OP assumes that $C$ is integral, so I answered in this setting. Mumford has a much weaker assumption which eliminates the kind of example you give. $\endgroup$– abxCommented Aug 27, 2014 at 13:08
-