I am currently reading about line bundles and have come across the following:
- The twisting sheaf of Serre, $\mathscr{O}(1).$
- To twist a sheaf $\mathscr{F}$ (of $\mathscr{O}_X$ modules) by $n,$ or $\mathscr{O}(n)$ means to tensor $\mathscr{F}$ by $\mathscr{O}(n)$ to get $\mathscr{F} \otimes \mathscr{O}(n).$
- More generally, to twist $\mathscr{F}$ by a line bundle $\mathscr{L}$ we tensor by $\mathscr{L}$ to get $\mathscr{F} \otimes \mathscr{L}.$
I was wondering where the language of "twisting" comes from. Is there perhaps a geometric reason? Or a historical one? In what sense is $\mathscr{F}$ twisted by $\mathscr{L}$?. Why does the Serre twisting sheaf have its name? I've looked in both Vakil and Hartshorne, and haven't found a motivation for the terminology.
