Understanding the Dimension of a Variety in Algebraic Geometry

[Newtime]

This should be a simple question for anyone familiar with basic algebraic geometry, but the concept is getting the best of me. Basically I am unclear on the concept of dimension of a variety. I'm still stuck in the algebraic mindset that dimension is the (minimum) number of elements needed to span the space, but that doesn't seem to carry over. I have several books on the subject, and the most intuitive definition I've found was that the dimension of a variety is the smallest r such that the variety is isomorphic to projective r-space. However, this still has me confused: how does one compute/find/understand the dimension of a projective space? I know projective n space has a defined dimension, but I can't understand why...

In particular, one knows a smooth point of a variety is a point at which the dimension of the (affine) tangent space is locally constant. I can translate this back into the original vector space and see the dimension, but I don't see how this carries over into projective space.

I suppose I'm looking for a little intuition in this matter...

Also, I'm posting this in the "Topology and Geometry" forum also. Mods, if this is contrary to any rules, please remove whichever posting is in the wrong place, but since this is an algebraic geometry question, I figured it would be best to get both perspectives.

Thanks in advance.

 

[Petr Mugver]

A variety is a topological space with an atlas, that is, a set of charts, that is, a set of invertible homeomorphisms from the variety to R^n, with certain properties (for more detail, see any book on differential geometry, I recommend you the one by Flanders, or a more difficult one, Spivak). In easy and a bit improper words, a variety is a set which has n coordinates... n is called the dimension of the variety.

 

[Newtime]

A variety is a topological space with an atlas, that is, a set of charts, that is, a set of invertible homeomorphisms from the variety to R^n, with certain properties (for more detail, see any book on differential geometry, I recommend you the one by Flanders, or a more difficult one, Spivak). In easy and a bit improper words, a variety is a set which has n coordinates... n is called the dimension of the variety.

So what you're saying is similar to the notion of dimension in linear algebra. Informally, the dimension of a variety is the number of parameters needed to describe it locally? Or the number of "directions" we can go if we want to choose some point? I'll also have a look at the book you mentioned. Thanks for your reply.

 

[adriank]

A variety is a topological space with an atlas, that is, a set of charts, that is, a set of invertible homeomorphisms from the variety to R^n, with certain properties (for more detail, see any book on differential geometry, I recommend you the one by Flanders, or a more difficult one, Spivak). In easy and a bit improper words, a variety is a set which has n coordinates... n is called the dimension of the variety.

There's some confusion in terminology. In many non-English languages (such as French), a cognate of the word "variety" is used to mean "manifold". In English, "variety" pretty much means "algebraic variety", which is a zero set of some set of polynomials.

I guess the usual definition is that if X is an irreducible algebraic variety, then dim(X) is the transcendence degree of the field k(X) of rational functions on X over k. That is, if dim(X) = n, then there are n algebraically independent rational functions on X, and any n+1 rational functions on X are algebraically dependent. (Elements f₁,...,f_r of k(X) are algebraically independent over k if and only if, given a polynomial P on r variables with coefficients in k, P(f₁,...,f_r) = 0 implies P = 0.) So this definition is pretty similar to the linear algebra definition of dimension, where the only real difference is algebraic independence vs. linear independence.

(By the way, this thread probably properly belongs in the Topology and Geometry forum, but since the discussion is here, let's keep it here.)

 

[Petr Mugver]

There's some confusion in terminology. In many non-English languages (such as French), a cognate of the word "variety" is used to mean "manifold". In English, "variety" pretty much means "algebraic variety", which is a zero set of some set of polynomials.

Uhm you are right, I was talking about manifolds, and probably Newtime was talking about what you say (about which I know exactly nothing). I apologize: the confusion you mentioned about French also applies to italian! (my language)

 

[Newtime]

There's some confusion in terminology. In many non-English languages (such as French), a cognate of the word "variety" is used to mean "manifold". In English, "variety" pretty much means "algebraic variety", which is a zero set of some set of polynomials.

I guess the usual definition is that if X is an irreducible algebraic variety, then dim(X) is the transcendence degree of the field k(X) of rational functions on X over k. That is, if dim(X) = n, then there are n algebraically independent rational functions on X, and any n+1 rational functions on X are algebraically dependent. (Elements f₁,...,f_r of k(X) are algebraically independent over k if and only if, given a polynomial P on r variables with coefficients in k, P(f₁,...,f_r) = 0 implies P = 0.) So this definition is pretty similar to the linear algebra definition of dimension, where the only real difference is algebraic independence vs. linear independence.

(By the way, this thread probably properly belongs in the Topology and Geometry forum, but since the discussion is here, let's keep it here.)

Thanks for the reply. This looks like a good way of thinking of it.

I posted it in the topology and geometry section as well, but there are still no replies.