# Good problems in Algebraic Geometry

I am now using Fulton's book Algebraic Curves to learn algebraic geometry from and have just finished chapter 2. However I feel that the problems are not very inspiring (at the moment at least) and lack some depth. Where is a good source of problems in algebraic geometry that I can find at least at the level of Fulton?

I don't mind if people recommend specific problems from Hartshorne say as long as I can do them with the tools I have from Fulton. To be more specific, the next chapter of Fulton is on local properties of plane curves and computing intersection numbers, so if one can recommend problems for these, it would be good.

Thanks.

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Have you seen Shafarevich's "basic algebraic geometry I " and AG by Daniel Perrin? – Ehsan M. Kermani Mar 10 '13 at 11:21
For beginners Shafarevich might be a little rough, imo. Try Reid's "Undergraduate Algebraic Geometry". It has drawings, galore examples and good exposition. – DonAntonio Mar 10 '13 at 11:37
@EhsanM.Kermani I am taking a look at Perrin's book right now. – user38268 Mar 10 '13 at 11:46

Evan Bullock's page for his Introduction to Algebraic Geometry might interest you.

Apart from 11 homeworks containing 7 or 8 exercises each, you'll find on it several interesting hand-outs and excerpts on Veronese, Segre, Grassmannians, symmetric and alternating tensors, dimension, ... by him or outstanding mathematicians like Harris, Atiyah-Mcdonald and others.
Bullock's course has as accompanying textbook Shafarevich's Basic Algebraic Geometry , which is at about the same level as Fulton: both are favourite introductions to algebraic geometry.

The exercises fit beautifully with Fulton. Here is an example:
In exercise 2.34 Fulton asks you to show that the sum $f=f_{d-1}+f_d$ of two homogeneous polynomials of degrees $d-1,d$ in $n$ variables (without common factors) is irreducible.
And Bullock in his first mid-term examination asks you to show that the zero locus $V(f)\subset \mathbb A^{n}$ of $f$ is birational to $\mathbb A^{n-1}$.
The icing on that particular cake is that you will find a solution to this exercise (and to others) here.

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Dear Georges, thanks for your answer. A user about recommended the book of Perrin, I skimmed through the first few chapters and the exercises look challenging. What do you think of this book? The thing with Fulton is, I spoke to my supervisor and he agrees with me that the exercises are not too inspiring. – user38268 Mar 10 '13 at 13:11
Dear Benja: I think that Perrin's book is just fantastic! It is written in the language of classical algebraic geometry but opens some vistas on scheme theory, for example when discussing the algebraic geometry of zero-dimensional algebraic subsets of $\mathbb A^n$. Fulton's decision to remain deliberately elementary has as consequence that you have to ingurgitate technical algebraic lemmas like Proposition 6 in Chapter 2, which I remember really baffled me as a beginner . This is of course quite ironic, given Fulton's egregious command of scheme theory! (to be continued) – Georges Elencwajg Mar 10 '13 at 13:47
(continued) Perrin introduces Čech cohomology, but makes some very judicious choices, like proving the vanishing of cohomology for quasi-coherent sheaves on affine varieties only for $\check H^1$. All in all, his is a remarkable book that I can't recommend too much (I didn't mention it because I don't think of it primarily as a book of exercises). – Georges Elencwajg Mar 10 '13 at 13:48
Thanks for your kind words. I will speak to my supervisor tomorrow. I was reading Proposition 6 in Chapter 2 and I too was stunned with this proof. Do you know a cleaner version of this proof? I don't mind if it uses some more advanced methods. If you visit my profile you'll see that I like commutative algebra! – user38268 Mar 10 '13 at 13:56
And on a lighter note, I owe you a full disclosure: Perrin is my mathematical brother! We had the same Ph.D. advisor, André Hirschowitz, but strangely I have never had any mathematical contact with him and have only seen him two of three times in my life. That was enough though to realize that he is a very nice person, très sympathique ! – Georges Elencwajg Mar 10 '13 at 13:57

Thomas A. Garrity has led a remarkable project aiming at teaching elementary algebraic geometry by means of problems.
Definitions are given but there are no proofs. The readers, meticulously guided by a progression of exercises which constitute the heart of the course, should find the proofs by themselves: the Moore method in all its splendour!
The level is more elementary than Fulton's.

This project has very recently become the AMS book Algebraic Geometry: A Problem Solving Approach.
I haven't seen that book, but I believe that my first link above is to the manuscript used for the published book.

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