# Are good introductory/pedagogical problems in algebraic geometry rare?

I have just started reading Elementary Algebraic Geometry by Hulek. It is a nice book but I find that it doesn't give many problems (about 10 to 15 per chapter), and that the exercises present are a bit boring (mainly specific case work, seemingly arbitrary curves, etc.).

This is in stark contrast with say, Modern Graph Theory by Bollobas: plenty of fun problems.

A friend told me he had experienced the same with other Algebraic Geometry books. And my lecturer told us that this might be related to the nature of algebraic geometry. Indeed, so much theory is needed before being able to properly analyze the most basic problems...

Thoughts? Counter-examples (i.e. introductory books with many fun problem)?

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This should perhaps be community wiki? – Scott Morrison Oct 23 '09 at 3:17
Done, thank you. – Randomblue Oct 23 '09 at 8:54

I highly recommend Fultons book "Algebraic curves" It's available on his webpage It's a very good introduction, and in the first chapter there are 54 exercises.

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I'd also recommend Fulton's book "Algebraic curves". I recently used the first few chapters to teach an intro course to algebraic geometry and the students tended to like it. Two other good intro books might be Karen E. Smith's "An Invitation to Algebraic Geometry", or Harris's "Algebraic Geometry: A First Course".

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also Shafarevich; Basic algebraic geometry – roy smith May 12 '11 at 20:50

Eisenbud & Harris' book "The Geometry of Schemes" & Ravi Vakil's lecture notes both have some good problems in them.

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I think Cox-Little-O'Shea's Ideals, Varieties, and Algorithms is a great text if you're looking at algebraic geometry for the first time. It focuses on the computational side of algebraic geometry and the commutative algebra behind it, so you won't hear about schemes, cohomology, etc., but it helps develop your intuition for algebraic geometry problems and skills in commutative algebra, and has loads of fun problems in every chapter. In particular, the chapter on projective algebraic geometry is awesome and has tons of interesting, instructive, and approachable exercises.

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Hartshorne has lots of problems and while a lot of them I probably wouldn't say are fun there are certainly a lot of them which are worth doing.

But now that I have given the standard answer I can't actually think of any others and it just occurred to me that Hartshorne might not be the sort of introductory text you are looking for?

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I don't know how elementary this is, but Igor Dolgachev's online classical algebraic geometry book contains many exercises, as does the first chapter of "Geometry of algebraic ruves" (this whole book is just a bunch of exercises, but only the first chapter can be called elementary).

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It may be true that most books start with commutative algebra, but I think that it's possible to teach alggeom with simple pedagogical examples. My favorite one is about the definition of a point for rings like K x L where K and L are fields - there's no distinction between prime and maximal ideals and that simplifies things a lot.

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By comparing with graph theory, I think you are setting algebraic geometry up to fail. Graph theory is a subject where it is easy to make appealing exercises. I know, because I taught such a course twice in a row, and had no difficulty making problems that I thought interesting, and that the students also found interesting and moreover doable; the course was a joy to teach. Algebraic geometry isn't my field, so I have no particular books with good exercises to suggest. In number theory there are several collections of problems of high quality, for example Sierpinski's, but I don't know of any in algebraic geometry. Market opportunity?

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