Learning projective geometry

My ultimate goal is to learn some algebraic geometry with the more concrete immediate goal of understanding things like how $\mathbb R\mathrm P^2$ is embedded in $\mathbb C\mathrm P^2$ or how $\mathbb C\mathrm P^2$ can be realised as the symmetric product of $\mathbb C\mathrm P^1$.

It seems to me that a necessary prerequisite is a solid understanding of projective geometry.

My problem is that I find it hard to develop some intuition about it: I keep mentally translating complex lines into real planes and try to visualise it - but it is not getting me very far...

I have been reading these notes. While I have no troubles following them, I cannot really see things like two projective lines meet at a point.

So my question is: what would you recommend as far as the way of thinking about projective spaces and geometry is concerned? Should I first develop more confidence with computations and proofs and build intuition on that? Should I stop translating from complex to real coordinates?

Solution 1:

I feel that the thing to do is to start off by thinking about real projective spaces. There are also some tricks to help understand projective spaces in general.

One trick is to "lift" back to the underlying vector space. For example, to come to grips with the idea that any two projective lines in $\mathbb P^2(\mathbb R)$ meet in one point, we can think about what happens in the vector space $\mathbb R^3.$ A projective line is a linear collection of projective points, which themselves are described by lines through the origin in $\mathbb R^3.$ So, a projective line is described by a plane through the origin in $\mathbb R^3.$ Translating what "two lines in $\mathbb P^2(\mathbb R)$ meet in a point" means to their representatives in $\mathbb R^3,$ the statement becomes that any two planes through the origin intersect in a unique line. Can you visualize this?

Another trick is to remember that projective space decomposes (non-uniquely) into disjoint pieces: $\mathbb P^n = \mathbb A^n \sqcup\mathbb P^{n-1}.$ Often times, we can ignore the fact that we're working with projective space, if the behaviour we're interested in takes place entirely within the $\mathbb A^n$ piece.

For example, the projective plane is what we get by adding the projective line to the affine plane $\mathbb A^2$ (in our case $\mathbb A^2 = \mathbb R^2,$ but without the vector space "structure", i.e., we don't add points of affine space, or multiply them by scalars). Remember from your notes, the projective line $\mathbb P^1(\mathbb R)$ can be thought of as the circle $S^1$ with antipodal points identified. If we put $S^1$ inside $\mathbb R^2$ as the unit circle centred at the origin, then every line through the origin defines a unique point of $\mathbb P^1,$ by its intersection with $S^1.$ This means that the projective plane is what we get by adding to the affine plane one point for each direction (i.e. slope!) that a line can have in $\mathbb R^2.$ If we are interested in the intersection of two lines in $\mathbb R^2$ that we know have different slopes, then we can ignore projective geometry. But, if two lines are parallel, then they do not intersect in $\mathbb R^2,$ but they do define the same point of $\mathbb P^1,$ which means that they intersect in the $\mathbb P^1$ portion of $\mathbb P^2$ ("at infinity").

Regarding visualizing complex geometry as real geometry, well, it's limited, because visualizing geometry past 4 dimensions isn't really possible. So, as soon as we think about spaces of complex dimension two or more, we're stuck. I think it's better to understand the real picture where we can, and then try to accept that the complex numbers arise in a way that preserves (and simplifies) the essential algebraic properties of the real numbers. When you study algebraic geometry, you will learn that geometric properties (in the situations we can visualize) can be expressed by algebraic conditions (vanishing of polynomials, etc.). The algebraic conditions can make perfect sense over different fields and in any dimension, but you will still only be able to visualize the geometry in $\mathbb R^1,\mathbb R^2,\mathbb R^3$ (maybe also over finite fields...). This doesn't mean we cannot draw diagrams to help our understanding, but those diagrams will no longer be so closely linked to the literal solutions to our equations, if that makes sense. But it's surprising how far one can get with "meaningless" pictures, so I encourage you to draw them in the real case, and to keep them in mind over other fields!

Anyway, those notes you are reading look very nice, I would certainly recommend you to keep reading them, and things will become clearer the more you do.