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I'm a physicist attempting to learn twistor theory. I'm confused by this notion of a tautological line bundle. So far, the most accessible source has been http://en.wikipedia.org/wiki/Tautological_line_bundle" [Broken]. In the definition, they say v in x. Do they mean a point in RP^n again? If so, why didn't they write RP^n x RP^n in the first place?

So, what I'm getting out of the second paragraph is RP^n is the base space, and we're adding a line acting as a fiber at every point in it. More specifically, we're adding the line that passes through the point. If so, why do they say RP^n x R^(n+1) instead of RP^n x R^n in the definition? Regardless, it seems that after doing this, my total space will just be R^(n+1). This statement, however, disagrees with their claim at the bottom that I'll recover a Mobius strip for n=1.

In twistor theory, I'm primarily concerned with "O(1), the dual of the tautological line bundle O(-1) over CP^1". Where does this O(-1) notation come from? Is it part of some more exciting area of math?

Thanks in advance! Chris

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# Tautological line bundle?

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