Quick question: Tensoring a 2-torsion line bundle with a rank 2 nonsplit extension over a curve of genus 1

Question

Everything is complex algebraic.

Over a curve $C$ of genus $1$, let $V$ be a rank 2 vector bundle with $\deg \det(V)=1$, which is a nonsplit extension of $\mathcal{O}_C$ and $\mathcal{O}_C(p)$, where $p$ is a point on $C$. I'd like to show that if $L$ is a line bundle such that $V\otimes L\simeq V$ then $L^{\otimes 2}\simeq \mathcal{O}_C$.

Apparently, $\deg(L)=0$ by Chern class. So I only need $h^0(L^{\otimes 2})=1$ to finish. Please help. Thank you very much.

Answer 1

In general, given two vector bundles $E,F$ of ranks $e,f$ on any variety $C$ we have the formula: $$\operatorname {det} (E\otimes F)= (\operatorname {det} E)^{\otimes f} \otimes (\operatorname {det} F)^{\otimes e} $$ where $\operatorname {det} E=\wedge^eE$ etc.

Applying this to the case at hand, we write $$ \operatorname {det}V \stackrel {\text{hyp}}{=} \operatorname {det}(V\otimes L) = \operatorname {det}V \otimes L^{\otimes 2} $$ so that by cancelling $ \operatorname {det}V$ we get the desired equality $$\mathcal O=L^{\otimes 2}$$