What shall I write for a reason for applying graduate school for algebraic geometry? I'm a undergraduate applying a graduate school this year and now I'm writing a letter of self-introduction.
To be honest, I don't know what exactly is algebraic geometry and I think 99% of undergraduates don't know what it is in general since it requires so many preliminaries.
I'm applying for algebraic geometry, since I know that this is a very deep and intrinsic subject which is itself important and its technique can be used to prove other parts of mathematics. For an example, Fermat's last theorem was proven by algebraic geometry method.
Moreover, I want to feel and touch Grothendieck's arguments and ideas so that I can really say how great he it.
But, that is it and I think professors won't see this nice.
Please help me what should I write for the reason applying for algebraic geometry.. This question sounds stupid, but it is really what I have in my mind now and I really want to get to study algebraic geometry..
 A: Here are some personal thoughts (speaking as an algebraic geometer, and someone who reads such statements.) I hope they are helpful.


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*As user148177 says, "don't BS". (In case it is not clear to non-native speakers of English, "BS" is an abbreviation for "bullshit", meaning to pretend to know more than you do.) Nobody expects you to know much about algebraic geometry, much less know what is in "in general", or to know all the preliminaries. If that standard was applied to applicants, nobody would ever start a PhD (in any subject)! Don't try to impress anyone with fancy notions; be honest! (I recently read an application that had a long "research statement" full of fancy notions; it was immediately evident that it had been copied from the internet.)

*You say you want "to feel and touch Grothendieck's arguments and ideas so that I can really say how great he i[s]." I understand this motivation, but some readers might not appreciate the sentiment so much. It might sound like you are too fixated on the work of one (albeit great) individual: for instance, what will you do if your advisor wants you to work on something not so close to Grothendieck's kind of algebraic geometry? Will you be flexible and adapt, or run away to the library and bury your head in EGA for three years? It's good to talk about your admiration for Grothendieck, but remember that doing a PhD isn't just about learning from the greats: you have to make your own contribution. Your letter should show that you understand that, even if you don't know what that contribution will be.

*Again echoing user148177, I advise you to try to say something specific and concrete that appeals to you about algebraic geometry. Have you looked at the introductory textbooks, like Reid and Shafarevich? Was there a topic or result in there that you liked, even if you didn't understand it fully? Talk about that, in a straightforward way. If your letter shows that you have no clear idea about the basic ideas and techniques of algebraic geometry, a potential advisor will (justifiably) wonder what is going to happen when you have to start learning these. Put another way, you have to demonstrate that you really do have some enthusiasm for algebraic geometry (even if you don't know much), rather than the idea of algebraic geometry.
Let me finish by saying that your passion is a great asset to you, and I don't mean to dampen it at all. If you can convey the same passion while addressing the points here and in the comments, your letter will be in great shape. 
Good luck!
