# Should I study projective geometry or commutative algebra as prerequisite to start algebraic geometry?

I am looking to study Algebraic Geometry but some books list projective geometry as a prerequisite and some list commutative algebra.

I have taken one semester of abstract algebra, real analysis, complex analysis, topology, combinatorics, and differential geometry.

I have not taken a course in projective geometry nor commutative algebra.

Which would be more important as a prerequisite if I want to start learning algebraic geometry?

Do you have any books to recommend?

• Knowledge of commutative rings/algebra is more or less a necessity, I believe. Nov 22, 2016 at 5:14
• Yes, skip projective geometry and learn commutative algebra. Nov 22, 2016 at 5:36

I think Commutativa Algebra is more important, yet there are very good books (e.g. Klaus Hulek's Elementary Algebraic Geometry, Fulton's Algebraic Curves) that introduce the necessary concepts and results from commutative algebra as they are needed. So If you don't really care about Commutative Algebra itself (or its applications in other areas like algebraic number theory), I suggest you start already with introductory algebraic geometry (Reid's Undergraduate Algebraic Geometry is another good reference).

Projective geometry is something you will definitely learn along the way.

Marco Flores has given a very good answer.

I would like to echo one aspect of his answer to suggest that if one wants to learn algebraic geometry, one can just start learning it. There are several good books available beginning at an undergraduate level, such as Reid's book that Marco Flores mentions. Later on, one can choose various approaches: the commutative algebra approach of Hartshorne's book, the differential geometry/topology approach of Griffiths and Harris, the related (but more bare-bones) approach of Mumford's book on projective varieties, ... . There are lots of "road map" questions about algebraic geometry texts here on Math.SE and also on MO that you can consult for guidance.

Finally, let me mention two answers on MO that push against the idea that algebraic geometry should be regarded as a branch of commutative algebra: this one and this one. I agree with their sentiment.

• (For OP) I think the following is helpful to think about with regards to tracing's last point. When one studies the theory of manifolds a non-trivial portion of it is of an analytic nature. That said, when one is really doing the subject it doesn't feel analytic. One often times reduces some question to some analytic result which one then cites, but you don't overly concern yourself with that. In fact, you let the geometry dictate how you understand the analysis. This is why people can read a book on complex manifolds with knowing almost zero several complex variables. Anyways, this is Nov 23, 2016 at 14:03
• to me how the algebraic geometry to commutative algebra situation feels. One uses commutative algebra constantly (it's the local study of schemes) but I don't think about it that way--in fact I think about the commutative algebra in terms of the geometry. Most of the commutative algebra I've learned has been on-the-fly when it was needed in my studies of AG. As an anecdote I have a friend who does intersection theory for a living, and often times forgets what the Nullstellsatz says (it's so obvious from a geometric perspective!). Another anecdote, is that I've never needed associated primes. Nov 23, 2016 at 14:06