Prerequisites For Algebraic Geometry?

[qspeechc]

Hi everyone.

What topics are prerequisites for algebraic geometry, at the undergrad level?

Obviously abstract algebra... commutative algebra? (What is that anyway?) Is differential geometry required? What are the prerequisites beside the usual "mathematical maturity"?

Thanks.

 

[Klockan3]

I think it is just a normal abstract algebra course and knowledge of the basics of maths. Commutative algebra of course studies commutative algebras! Commutative means that all elements commute. Two elements commute under the operation + if a+b=b+a.

 

[qspeechc]

Ok. But then why is it usual that one does differential geometry before algebraic geometry? Isn't DG a prereq to AG?

So algebra is the only prereq? Really? No analysis (real/complex), DG, linear algebra,...?

 

[rasmhop]

Depends on the course. An undergraduate course may teach you the necessary commutative algebra as you go along and skip some generality to keep the prerequisites down. For a full general course in algebraic geometry using something like Hartshorne's algebraic geometry you will need a strong background in both algebra and commutative algebra (the theory of commutative rings). For instance the 800 page book Commutative algebra with a view toward algebraic geometry was largely written to give students the appropriate background in commutative algebra to tackle Hartshorne. I can't imagine such a course at the undergrad level, but you may well have something in between. The prerequisites depend entirely on how algebraic geometry is presented. For instance Ideal, Varieties and Algorithms is a very elementary introduction to algebraic geometry that barely even require much abstract algebra. Check your course catalog, it probably lists the prerequisites. If it lists differential geometry, then it's a prerequisite. If this is self-study then read the introduction to your book and it should mention the prerequisites.

By the way when you require algebra it's usually implied that linear algebra is also required.

 

[qspeechc]

Oh, so it depends on the way the course is presented. The handbook, btw, doesn't give the prereqs for some courses, oddly. They want us to start speaking with lecturers and mingling with faculty, and this is in that effort. Bloody terrible, because I try to avoid mingling with people wherever I can; hence why I asked the question here.

I suppose now I'll have to e-mail the lecturer or go speak with him. *Sigh*. Well, thanks for the responses.

 

[Dembadon]

Hi everyone.

What topics are prerequisites for algebraic geometry, at the undergrad level?

Obviously abstract algebra... commutative algebra? (What is that anyway?) Is differential geometry required? What are the prerequisites beside the usual "mathematical maturity"?

Thanks.

What course(s) does your university list as the prequisite(s) on the course description for the algebraic geometry class?

--------------------------------------------

Edit: Here is some info from http://en.wikipedia.org/wiki/Algebraic_geometry" which may help you as well (Underlining is mine). You may also want to talk to someone in the math department at your university.

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of polynomial equations in many variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations, as to find some solution; this leads into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

The fundamental objects of study in algebraic geometry are algebraic varieties, geometric manifestations of solutions of systems of polynomial equations. Plane algebraic curves, which include lines, circles, parabolas, lemniscates, and Cassini ovals, form one of the best studied classes of algebraic varieties. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve relative position of different curves and relations between the curves given by different equations.

 

[Ivpoly]

Technically there are not many formal prerequisites, in fact I think a lot of books claim to start from scratch. The bible (Hartshorne) only formally requires Atiyah Mcdonald.

However to get the right intuition for Alg Geom in hindsight I think the best way to approach it is to try and know some stuff about Manifolds, then read the book "An Invitation to Algebraic Geometry".

The reason is eventually there are stuff with "sheafs" and "schemes" and without something concrete to latch onto, the definitions will feel like horsepills and there will be plenty.

 

[qspeechc]

Ok, thanks everyone. I'll pop into the the lecturer's office on Monday, or drop him an e-mail. This is an undergrad course, so I doubt the prereqs will be too demanding. I won't have time to learn about manifolds if that's a prereq. Otherwise I have done abstract algebra, real analysis, and complex analysis; which I hope is enough.

 

[MathematicalPhysicist]

Oh, so it depends on the way the course is presented. The handbook, btw, doesn't give the prereqs for some courses, oddly. They want us to start speaking with lecturers and mingling with faculty, and this is in that effort. Bloody terrible, because I try to avoid mingling with people wherever I can; hence why I asked the question here.

I suppose now I'll have to e-mail the lecturer or go speak with him. *Sigh*. Well, thanks for the responses.

Well, if you plan for grad school or even outside academia you should know how to mingle.

 

[qspeechc]

Well, if you plan for grad school or even outside academia you should know how to mingle.

I know. People keep telling me this. But that's just the way I am: a very un-sociable person. I'm not putting it on, and I can't change the way I am. I find it very difficult to mingle. And why should I torture myself trying to fit into other people's ideas of how I should be.
But you are correct, I do need to open up.

 

[qspeechc]

By the way, what is this book you mention here:

However to get the right intuition for Alg Geom in hindsight I think the best way to approach it is to try and know some stuff about Manifolds, then read the book "An Invitation to Algebraic Geometry".

 

[Ivpoly]

https://www.amazon.com/dp/0387989803/?tag=pfamazon01-20

They take for granted some things about manifolds, (thats from differential geometry)

They review a lot of the basic commutative algebra making it "ideal" for someone cutting their teeth.

There is another good book mentioned above by rasmhop "Ideal, Varieties, Algorithms" by Cox, Little and Oshea.

As mentioned, that book does not have much prerequisites, they start from scratch.

 

[qspeechc]

Thank you for the recommendations. They both seem to be at about the correct level for me. I think the book used for the course is this one:

https://www.amazon.com/dp/0521691419/?tag=pfamazon01-20

since that's what they used in previous years, with the same lecturer.

 

[Dembadon]

I know. People keep telling me this. But that's just the way I am: a very un-sociable person. I'm not putting it on, and I can't change the way I am. I find it very difficult to mingle. And why should I torture myself trying to fit into other people's ideas of how I should be.
But you are correct, I do need to open up.

[off-topic] Socializing with others does not mean you have to compromise your identity. Not everyone has to like you, and not everyone will.

Also, please do not just accept that you are anti-social. An introvert must learn how to socialize. I agree with you, it is difficult, but that does not make it impossible nor something which one can afford leave undeveloped. You can change the way you are. Again, this is about a skill, and not your identity. Nobody is encouraging you to sacrifice your uniqueness, but rather develop an essential skill. The more you do it the more comfortable you'll become. Avoiding it is not wise. [/off-topic]