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I want to learn more about classical varieties. How do I proceed?

I was looking for more information on Classical Varieties and what exactly it means. I worked through the wikipedia article - http://en.wikipedia.org/wiki/Algebraic_variety, but I do not understand what they are used for.

I have taken calculus, multivariable calculus, linear algebra, ODEs, PDEs, (engineering math background). This sort of math has always been a mystery to me. What are the financial implications of such math? I have little experience in geometry (Is this geometry?) and would like to unravel a bit of the mystery, mostly for curiosity's sake. My goal in this endeavor is to understand a bit more of how collaboration, problem solving, and "engineering" is done in mathematics, as well as learn some interesting things (which I am assuming are not practically useful??)

A little nudging of "start with this" or "This should be at your level" or "Whoah you totally have the wrong idea, look at this" would be great

Thanks

edit:

From the amazon review of Undergraduate Algebraic Geometry - "

The style is friendly, straightforward and unpretentious. Everything is well motivated, and one occasionally gets to hear the author's personal perspective or view about a certain topic. I will quote two examples. When discussing the Zariski topology, the author writes "The Zariski topology may cause trouble to some students; since it is only being used as a language, and has almost no content, the difficulty is likely to be psychological rather than technical". This was very calming for me to read, as I have been previously struggling with the "deep meaning" of the Zariski topology, and no book has had the honesty to tell me that I shouldn't worry that much about it. As a second example of the author's style, after a Q.E.D. in page 53 the author explains that "The proof of (b) is a typical algebraist's proof: it's logically very neat, but almost completely hides the content: the real point is that ..."

"

I think I'll get this book thanks

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    Could you please elaborate? I would really appreciate it! :)2012-06-18

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