Space of algebraic curves of degree $d$ is compact

Question

In an article I am reading it states "the space of algebraic curves of a given degree $d$ is compact". It seems to take this as a basic fact, as there is no explanation on this. I was wondering could someone please explain what this means?

Answer 1

Let $\mathbb{K}$ denote $\mathbb{R}$ or $\mathbb{C}$.

The vector space of homogeneous polynomials in $\mathbb{K}[X,Y,Z]$ of degree $d$ is isomorphic to $\mathbb{K}^{ {d+2} \choose {2}}$. For example, if $d=2$ it is the span of $\{X^2,Y^2,Z^2,XY,XZ, YZ\}$. Now two homogeneous polynomials define the same algebraic curve if they are scalar multiples of each other. So the space of algebraic curves is basically $\mathbb{P}^{{{d+2} \choose {2}} -1}(\mathbb{K})$. Under the standard quotient topology both the complex and real projective space are compact spaces.

I am quite sure this is what they mean.