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?



Negative degree line bundles over a singular projective curve have no sections?
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 $\deg \mathcal{L}=\deg \mathrm{div}(s)\geq 0$.


