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 Post subject: FibersPosted: Wed, 14 Dec 2011 14:04:03 UTC
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I'm reading through the chapters in Algebraic Geometry, Harris, about Blow-ups and Resolving Singularities and he keeps mentioning fibers (for instance, in one example he says
the fiber over that point [the origin in ] is a copy of corresponding to the lines through that point), however, he doesn't seem to ever actually define what a fiber is, only something called the fiber product (which I assume is also something I will need).

So if someone could tell me what he means that would be much appreciated (I also noticed he never actually explains what a homogeneous equation is, which I know is a really basic concept (and we have subsequently covered it in lectures so it doesn't matter), but still!)

Thank you very much!

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 Post subject: Re: FibersPosted: Wed, 14 Dec 2011 14:49:44 UTC
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peccavi_2006 wrote:
I'm reading through the chapters in Algebraic Geometry, Harris, about Blow-ups and Resolving Singularities and he keeps mentioning fibers (for instance, in one example he says
the fiber over that point [the origin in ] is a copy of corresponding to the lines through that point), however, he doesn't seem to ever actually define what a fiber is, only something called the fiber product (which I assume is also something I will need).

So if someone could tell me what he means that would be much appreciated (I also noticed he never actually explains what a homogeneous equation is, which I know is a really basic concept (and we have subsequently covered it in lectures so it doesn't matter), but still!)

Thank you very much!

Suppose . The fibre over is in topology. However, in AG you need a bit more but that requires you to know about schemes (you have , and the fibre of over is ).

A homogeneous equation is exactly what you think it means --- the total degree of every term appearing in the equation is the same.

So you blow up at the origin, getting , where , where are the coordinates for and are the coordinates of . The fibre over any point of is still a point, but the fibre over (0,0) is the entire since there isn't any condition on the .

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 Post subject: Re: FibersPosted: Wed, 14 Dec 2011 15:30:04 UTC
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So if I had a point for some arbitrary , would the map be an isomorphism away from p, and at p, would that generalise to the fiber over p being ?

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 Post subject: Re: FibersPosted: Wed, 14 Dec 2011 16:14:05 UTC
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peccavi_2006 wrote:
So if I had a point for some arbitrary , would the map be an isomorphism away from p, and at p, would that generalise to the fiber over p being ?

If you are blowing up a point, yes. If you blow up along a subvariety you replace the subvariety by its projectivised normal bundle.

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 Post subject: Re: FibersPosted: Wed, 14 Dec 2011 17:25:35 UTC
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Excellent - thanks outermeasure

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