# Knots & Global sections of projective varieties

I’m trying to post something everyday. In any case, I spent a good chunk of time today editing a knot paper. Its not on the arxiv yet, but maybe I’ll put a link when it is. Anyways some math. The goal is to show if is projective over an algebraically closed field k then . The idea in Hartshorne is simple. Take and show it is integral over , hence constant as .

By identifying with some degree 0 element in some localization of it will be enough to show is integral over (once we have this we just take the degree 0 part). In particular we look at the standard affine patches and note . Also there is an isomorphism which descends from the isomorphism sending .

So now you identify with some fraction and verify for that . Recall is a degree zero, so considering guarantees than any monomial of degree N will cancel the denominator of . Consequently, . So in fact any power of times something of degree N is in . Use this to show . Then is finite over which shows is integral over (basic comm. alg. result). And that’s about it.

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- Published:
- May 15, 2009 / 11:38 am

- Category:
- alg. geo.

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