How should I think of lines and planes in projective space?

Question

I have been learning about projective varieties recently and I realised that I have some trouble trying to grasp what lines and planes are even in say $\Bbb{P}^3$. For one, how should I think about a line? Well given homogeneous coordinates for two points $A$ and $B$ I can write down an expression that gives all possible points on the "line between $A$ and $B$". However this does not seem satisfying because of the following.

In affine space, I am inclined to think of a line as a being a one dimensional $k$ - vector space that is isomorphic to $\Bbb{A}^1$ under an affine change of coordinates. A plane isomorphic to $k^2$, etc. Also, I am inclined to think that given two planes $P_1,P_2$ in $\Bbb{A}^3$ the "dimension" of their intersection is at most $1$.

However can all of this intuitive reasoning be transferred over to $\Bbb{P}^3$ say? By this I mean is the intersection of two "planes in $\Bbb{P}^3$" a "line"? This is of course assuming that we have defined a "plane" and "line" in $\Bbb{P}^3$. What should be the definition of a "plane" in $\Bbb{P}^3$?

Also, how many "hyperplanes" do I need to describe a line in $\Bbb{P}^n$?

I guess all this is confusing for me because my background has mainly been in a lot of algebra and not much on questions like that.

Answer 1

For terminology, see Hartshorne, exercise 2.11: In $\mathbb{P}^n$, a linear variety is defined as a variety given by linear polynomials. An equivalent (this needs checking!) definition is that it is the intersection of some hyperplanes (where a hyperplane is defined as the zero locus of a single linear polynomial).

Also you can check out

Is a 2-dimensional subspace always called a plane no matter what the dimensions of the space is?

For the visualization of projective space, see my answer to

Why the emphasis on Projective Space in Algebraic Geometry?

Now why am i telling you all this? Because, if one takes affine charts in projective space, a projective line becomes exactly what you describe: a line in a vector space (but with Zariski topology instead of the Euclidian). The same holds for a plane.

I would recommend you to check this out for yourself.

In general, this is how one can think of these things: reduce to affine charts. Projective space $\mathbb{P}^n$ is not that difficult, given that affine space $\mathbb{A}^n$ sits in it as a dense open subset, and its complement is "the points at infinity" and is isomorphic to $\mathbb{P}^{n-1}$. So when you reduce to affine charts, you barely lose information, and you can use your intuition.

I hope this answers all of your questions, as well as giving you an idea where to look when new questions pop op. Feel free to comment in case you need more information.

P.S. Sorry for commenting everywhere first, i was not planning on posting an answer..

Answer 2

You should think of projective varieties (say over $\mathbb{C}$) pretty much just as you do affine varieties except for the fact that they are compact, or better yet that no points of your variety can run off towards infinity. After all, projective varieties are obtained by gluing together affine varieties. Moreover, one of the reasons that we like to work in projective space is that it is a better place in which to discuss how varieties intersect. For example in $\mathbb{P}^2$ an irreducible curve of degree $d$ and an irreducible curve of degree $e$ will always intersect in $de$ points, as long as the curves are distinct and you count their points of intersection along with their multiplicity. Generally speaking (there are anomolous cases), two projective subvarieties in $\mathbb{P}^n$ of codimensions (codimension$=n-$dimension of the variety) say $p$ and $q$ will intersect in a subvariety of codimension $p+q$, thus intersecting varieties is kind of like multiplication of polynomials (google "Chow ring" for more on this). Thus two planes (which have codimension 1 in $\mathbb{P}^3$) in $\mathbb{P}^3$ will always intersect in a line (codimension $1+1=2$) as long as they are distinct. However for example in $\mathbb{P}^4$ two planes will intersect in a single point. Even more generally if you have $n$ (general) hypersurfaces in $\mathbb{P}^n$ then they will intersect in a codimension $n$ subvariety, which is a dimension zero and so just consists of points, which when counted with multiplicity is equal to the product of the degrees of the hypersurfaces you are intersecting.