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I am teaching an introductory course in algebraic geometry for masters and it turns out that many of them are not at all familiar with the notion of projective space. So it is necessary to spend two-three hours on explaining what this notion. I would like to know if there is a short (say 5-10 pages) carefully written note with a nice explanation of what is projective space containing some non-tautological statements (like Desargues theorem) that would help to students to get a feeling.

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I'm actually a masters student who is just now learning projective space on my own, and I found the introduction to projective space given in the book Ideals, Quotients, and Algorithms to be really good and eminently understandable, you might take a look at it and write some notes up based on it, on your own. –  heat death Oct 24 '12 at 16:56
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Miles Reid has a nice introductory book called Undergraduate Algebraic Geometry, which has the goal of providing motivation for algebraic geometry to undergrads (& beginning masters students). I highly recommend it, along with his book Undergraduate Commutative Algebra. These don't satisfy your condition of being 5-10 pages, but they are certainly at the right level.

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Andrew thank you for your answer! I agree that the level of these books is the right one. I tried to use UAG first time I was teaching the course (2 years ago) but this did not quite work out. I think the book is not really so well written, at least for my taste. In fact I wrote an email to Miles asking about the book and he told me that he never thought a course following the book :) he usually gives this book for his students for self-study. So I wrote notes for myself combining various sources (Fulton, Borcherds, Silverman...) but I want to improve the basic bit on projective space. –  agleaner Oct 25 '12 at 23:39
Dear @agleaner, actually, my experience with Reid's books was for self-study/reading courses also, so I can appreciate what you say. It is interesting to know that he didn't use the book for his course! –  Andrew Oct 26 '12 at 2:52
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