Comments on: Introduction to Algebraic Geometry (1 in a series) http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/ Because my LiveJournal is too silly Mon, 15 Mar 2010 18:50:59 +0100 http://wordpress.org/?v=2.8.6 hourly 1 By: Michi http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/comment-page-1/#comment-82252 Michi Sat, 23 Feb 2008 16:08:21 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/#comment-82252 mwanahamisi: I'm glad you're interested in the subject. I cannot guarantee neither that I know the answers to any questions you might have nor that I'll have time to elaborate on request. But you're free to ask. mwanahamisi: I’m glad you’re interested in the subject. I cannot guarantee neither that I know the answers to any questions you might have nor that I’ll have time to elaborate on request. But you’re free to ask.

]]> By: mwanahamisi http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/comment-page-1/#comment-82238 mwanahamisi Sat, 23 Feb 2008 15:20:31 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/#comment-82238 I am maths student in tanzania, and you summary on algebraic geometry is attracting my interest in the subject. Allow me to contact you whenever I need elaboration on this complex area I am maths student in tanzania, and you summary on algebraic geometry is attracting my interest in the subject. Allow me to contact you whenever I need elaboration on this complex area

]]> By: Ben http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/comment-page-1/#comment-82127 Ben Sat, 23 Feb 2008 01:43:31 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/#comment-82127 Have you tried Mumford's Red Book? It's the best I've found..... Have you tried Mumford’s Red Book? It’s the best I’ve found…..

]]> By: Carnival of Mathematics 1000 « JD2718 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/comment-page-1/#comment-82123 Carnival of Mathematics 1000 « JD2718 Sat, 23 Feb 2008 01:00:51 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/#comment-82123 [...] Difakis (won the ACM Turing award). 21 - Michi finished his thesis, and found the time to post an introduction to algebraic geometry (in two [...] [...] Difakis (won the ACM Turing award). 21 – Michi finished his thesis, and found the time to post an introduction to algebraic geometry (in two [...]

]]> By: Michi http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/comment-page-1/#comment-81915 Michi Thu, 21 Feb 2008 22:33:16 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/#comment-81915 John: Sure, that -is- a completely reasonable definition. And one made by, for instance, Shafarevich's rather seminal work. However, regardless of what the Wikipedia says, the literature does show more definitions than this. For instance, Smith-Kahanpää-Kekäläinen-Traves does define an affine algebraic variety in my way; and I do recall having seen it that way elsewhere as well. It all really boils down to what you want the language you define to work out for. With my definition of varieties, closed sets in the Zariski topologies are precisely subvarieties of the affine space, and open sets are precisely complements thereof. In your language, distinction would have to be made between reducible and irreducible closed sets - which, I felt, made the exposition slightly more awkward. John: Sure, that -is- a completely reasonable definition. And one made by, for instance, Shafarevich’s rather seminal work.

However, regardless of what the Wikipedia says, the literature does show more definitions than this. For instance, Smith-Kahanpää-Kekäläinen-Traves does define an affine algebraic variety in my way; and I do recall having seen it that way elsewhere as well.

It all really boils down to what you want the language you define to work out for. With my definition of varieties, closed sets in the Zariski topologies are precisely subvarieties of the affine space, and open sets are precisely complements thereof. In your language, distinction would have to be made between reducible and irreducible closed sets – which, I felt, made the exposition slightly more awkward.

]]> By: John http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/comment-page-1/#comment-81914 John Thu, 21 Feb 2008 22:22:39 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/#comment-81914 I was always under the impression that a set of common solutions to polynomials is called an algebraic set. A variety is only an irreducible algebraic set. Wikipedia seems to agree with me... http://en.wikipedia.org/wiki/Algebraic_variety I was always under the impression that a set of common solutions to polynomials is called an algebraic set.

A variety is only an irreducible algebraic set.

Wikipedia seems to agree with me…
http://en.wikipedia.org/wiki/Algebraic_variety

]]> By: Charles http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/comment-page-1/#comment-81865 Charles Thu, 21 Feb 2008 18:19:27 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/#comment-81865 Ahh, well, no particular request. And if you're going to talk about schemes (especially in general) then our viewpoints will diverge, because I'm focusing on varieties (albeit abstract varieties with potentially embedded components). I wish you good luck, and I'm going to keep reading. Ahh, well, no particular request. And if you’re going to talk about schemes (especially in general) then our viewpoints will diverge, because I’m focusing on varieties (albeit abstract varieties with potentially embedded components). I wish you good luck, and I’m going to keep reading.

]]> By: Brent Yorgey http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/comment-page-1/#comment-81815 Brent Yorgey Thu, 21 Feb 2008 15:52:20 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/#comment-81815 Cool! I look forward to reading more. =) Cool! I look forward to reading more. =)

]]> By: Michi http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/comment-page-1/#comment-81764 Michi Thu, 21 Feb 2008 13:07:51 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/#comment-81764 Thanks Charles, I actually wasn't planning on talking particularly much about Gröbner bases - mainly because that's an aspect I actually DO understand. I was planning on building up a setting where heading into sheaves looks natural pretty swiftly, and then go on from there to discuss sheaves and schemes. However, I'm always happy to consider requests for material my audience wants me to cover. :) Thanks Charles,

I actually wasn’t planning on talking particularly much about Gröbner bases – mainly because that’s an aspect I actually DO understand. I was planning on building up a setting where heading into sheaves looks natural pretty swiftly, and then go on from there to discuss sheaves and schemes.

However, I’m always happy to consider requests for material my audience wants me to cover. :)

]]> By: Charles http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/comment-page-1/#comment-81762 Charles Thu, 21 Feb 2008 13:03:43 +0000 http://blog.mikael.johanssons.org/archive/2008/02/introduction-to-algebraic-geometry-1-in-a-series/#comment-81762 Good post. You're going a bit more into the guts of the theory, but you might be interested in my series of posts http://rigtriv.wordpress.com/category/algebraic-geometry/algebraic-geometry-from-the-beginning/ which is more about the intuition behind everything...and I should get back to that, as I've just introduced sheaves... Oh, and if you're going to talk about Groebner Bases, that'd be great, and I'll just link back to you whenever I want to do a computation. Good luck! Good post. You’re going a bit more into the guts of the theory, but you might be interested in my series of posts http://rigtriv.wordpress.com/category/algebraic-geometry/algebraic-geometry-from-the-beginning/ which is more about the intuition behind everything…and I should get back to that, as I’ve just introduced sheaves…

Oh, and if you’re going to talk about Groebner Bases, that’d be great, and I’ll just link back to you whenever I want to do a computation.

Good luck!

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