Global section of tensor product of line bundles on a curve

Solution 1:

I am not sure what you are looking for, but let me make a statement answering your question negatively.

Let $X$ be a smooth projective curve of genus $g$ and let $d$ be a positive integer with $d<g$. Then there exists a line bundle $L_1$ of degree $d$ with no sections and a line bundle $L_2$ of degree $-1$ (obviously has no sections) such that $L_1\otimes L_2$ has a nonzero section. If $d>1$, clearly $L_2$ is not the inverse of $L_1$.

For this, let $P_r=\operatorname{Pic}^r X$, the set of all line bundles of degree $r$. These are all isomorphic and has dimension $g$. Since $d<g$ and the set of line bundles of degree $d$ with a section has dimension atmost $d$, a general element of $P_d$ has no sections. Fix such an $L_1$. The multiplication map $P_0\to P_d$, $M\mapsto M\otimes L_1$ is an isomorphism, so there exists an $M$ such that $M\otimes L_1$ has a nonzero section. Fix this $M$. So, $M\otimes L_1=O(P_1+\cdots+P_d)$, for some points $P_i$. Now, take $L_2=M\otimes O(-P_1)$.