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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!

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Your guess is correct. The geometric (or function-theoretic) meaning is that sections of $\omega(D)$ are sections of $\omega$ that are allowed to have simple poles along $D$.

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    Well, I'm sorry I have finished all my chances to give you reputation! Thank you very much again!2012-05-09