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I came across the following sentences which I cannot understand:

"In general, let $E$ be a vector bundle on a curve $C$, consider a point $p \in C$, and let $\xi \subseteq E_p$ denote a 1-dimensional subspace. Denote $E(\xi)$ to be the sheaf of sections of $E$ with at most a simple pole at $p$ in direction $\xi$."

Should I read "sections" as "sections of the vector bundle $E$ over the open $C \setminus {p}$"?

And what does "in direction $\xi$" mean?

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1 Answer 1

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I don't know what the author means but here is my guess .

Consider the sky-scraper sheaf $Sky_p(E_p/\xi)$ with support $p$ and fiber $E_p/\xi$.
Define a subsheaf $F\subset E$ by the exactness of $0\to F\to E\to Sky_p(E_p/\xi)\to 0$.
My guess is that $E(\xi)=F\otimes_\mathcal O \mathcal O(p)$.
This means that if $z$ is a local coordinate for $C$ at $p$, sections of $E(\xi)$ near $p$ are of the form $\frac {1}{z}s$ where $s$ is a local section of $E$ satisfying $s(p)\in \xi$.

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