User morton - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:27:51Z http://mathoverflow.net/feeds/user/3017 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10738/projective-to-affine Projective to Affine? Morton 2010-01-04T21:43:52Z 2010-12-03T18:22:06Z <p>Perhaps a basic question... how does one study affine algebraic geometry via projective geometry? For example, suppose I have two affine varieties which I want to prove are isomorphic, would it help to look at the projective closures (assuming I don't know of any other method for proving they are isomorphic)? How does one go back to the affine case from the projective closure (or "projectivization")? Sorry if this sounds confusing.</p> <p>Thanks, Morton</p> <p>Edit: Thanks for the replies. Being new to AG let me try and rephrase my quandary: It seems the projective setting is the most convenient to study AG but if I want to study properties of affine varieties, how does one use results of projective varieties in the affine case? I know this sounds vague but it is a fundamental doubt I have. </p> http://mathoverflow.net/questions/10282/alternative-undergraduate-analysis-texts/11011#11011 Answer by Morton for Alternative Undergraduate Analysis Texts Morton 2010-01-07T04:47:36Z 2010-01-07T04:47:36Z <p>Gaughan's <a href="http://www.amazon.com/Introduction-Analysis-Contemporary-undergraduate-mathematics/dp/0818501723" rel="nofollow">book</a> and the <a href="http://www.amazon.com/Introduction-Real-Analysis-John-DePree/dp/0471853917" rel="nofollow">book</a> by Swartz and Depree are excellent for undergraduate Analysis. Swartz Depree also does the Gauge integral.</p> http://mathoverflow.net/questions/10738/projective-to-affine/10794#10794 Comment by Morton Morton 2010-01-07T03:34:26Z 2010-01-07T03:34:26Z Thanks Maharana and Charles, this really helps. I would ideally like to see some concrete examples if you can suggest a reference. I'll get my hands on Iitaka's book. I don't know much about the Mori program but I am curious what happens in the case of an affine threefold, say X. Suppose I embed X in a projective P and arrive at a &quot;final&quot; space X'=P'\D', how does X' compare with X? What all can go wrong? What sort of pathologies arise? Is there a specific reference for affine threefolds you can suggest? Any reference I see only seems to address projective threefolds, the affine case eludes me.