Intro to Projective Varieties

I will assume a basic familiarity with projective space from now on (I don’t think Ive covered it in any previous posts). For a quick recap and a guide to my notation, we can define projective space on a finite dimensional vector space V over k by defining a n equivalence relation uRv iff there is a non-zero scalar \lambda\in k such that u=\lambda v. Then \mathbb{P}(V)=\frac{\left( V\setminus\{0\}\right)}{ R}.

Review/facts: dim\mathbb{P}(V)=dim V -1. Also, the way we usually think is \mathbb{P}(\mathbb{C})\cong S^2. The complex projective space (line) is just the compactification of the plane and hence the Riemann sphere. The fact that this is a compact manifold is no coincidence, in fact, all projective spaces can be given the structure of a compact manifold…but maybe this should not be mentioned, since I want to put a different topology on it and talk about varieties.

So our field and the dimension of the vector space are in some sense more important than the vector space itself, so I’ll notate from now on \mathbb{P}(V) as \mathbb{P}_k^n. If we take \pi: V\setminus\{0\}\to \mathbb{P}(V) as the standard projection onto the quotient, we use the notation (x_0 : \ldots : x_n) for \pi((x_0, \ldots , x_n)). These are called homogeneous coordinates. Note that these are only well-defined up to scalar multiple.

I haven’t developed on this blog any good motivation to now switch to projective space, but there are some good reasons. At first, it seems to just make things more complicated, but really it simplifies things in the long run. Also, there are some nice properties that you should check. In the affine case, lines can either be parallel and never intersect, or they intersect somewhere. In the projective case, all lines intersect.

So now we just extend the same definitions from the affine case to the projective case, but we are careful to make sure everything is well-defined.

Since homogeneous coordinates are determined up to multiplication by a scalar, we need to make sure our polynomials can deal with this. So we call a polynomial homogeneous of degree d, if every monomial has degree d. i.e. f(x_0, \ldots, x_n)=\sum a_{b_0\cdots b_n}x_0^{b_0}\cdots x_n^{b_n} where b_0+\cdots b_n=d. So, we have well-defined zero sets of polynomials since we can pull scalar multiples out: f(\lambda x_0, \ldots , \lambda x_n)=\lambda^d f(x_0, \ldots, x_n), i.e. the variety V(f)=\{ (x_0: \ldots : x_n)\in \mathbb{P}_k^n : f(x_0, \ldots, x_n)=0\} is well-defined. Note: If d=2, then we call these quadratic forms.

We call a subset V\subset \mathbb{P}_k^n a “projective variety” if there is a set of homogeneous polynomials T\subset k[x_0, \ldots , x_n] such that V=V(T)=\{P\in \mathbb{P}_k^n : f(P)=0 ~ \forall f\in T\}.

I’ll let you digest that, and learn a little more about projective space if this was your first experience dabbling with it. Our future plans are to try to figure out which parts of the affine theory we developed carries over unscathed, which parts break down, and which parts get simplified.


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