My question is how to define the twisting of a sheaf $\mathcal{L}$ by a divisor $D$.
In specific I'm interested in the twisting of the canonical bundle $\omega$ of a non-compact Riemann surface $X$ by a divisor of points. (The points are the missing points of the compactification, in my case they are a finite number.)
My guess on the definition is the following. I take $D=p_1+...+ p_n$. On $X$ Weil and Cartier divisors are the same, so $D$ is also a Cartier divisor and it is well defined the associated sheaf $\mathcal{L}(D)$. Then I "define" the twisted canonical bundle by:
$\omega(D)=\omega \otimes \mathcal{L}(D)$.
Is my "definition" correct? If it is I don't really see its geometric meaning. If it is not can you please point out a reference for it?
Thank you!