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Divisor — line bundle correspondence in algebraic geometry

After reading some introductory texts on Algebraic geometry, it seems that Divisors and Bundles are treated the same? How are they related?

I always thought that Divisors were Hypersurfaces of some manifold (something inside the manifold) and Bundles something that is "stuck" onto the manifold (something outside of it), where does my thinking go wrong?

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FMN
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    Divisors and line bundles. If you want more bundles, you'll have to think about cycles! – Dylan Moreland May 24 '12 at 16:49
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    It often happens that an "inside" thing is equivalent to an "outside" thing. For example, a subset $S$ of a set $X$ is an internal thing (an injective function $S \to X$) but it is also an external thing (a function $X \to { 0, 1 }$). – Qiaochu Yuan May 24 '12 at 16:50
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    You should look at this: http://math.stackexchange.com/a/1943/19379 – M Turgeon May 24 '12 at 16:59
  • Ahh thanks for clarifying Dylan! Starting to make a bit more sense now. Would I be right to think that the Line Bundle corresponding to a divisor is like saying the line bundle specifies the "flow" of the hypersurface and that's how they are connected? – FMN May 24 '12 at 17:01
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    Ohhh didn't see this question, thank you M Turgeon! – FMN May 24 '12 at 17:02

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