I am learning about the degree of algebraic sets. I know the definition from Wikipedia, but it is not too clear to me what it is. Could someone possibly explain to me exactly what property the degree of an algebraic set does capture, or how I should think about it?
PS: This was part of another question of mine, Basic question regarding degrees of algebraic sets, but I thought that maybe it would be better to ask this separately.
Degree is best thought of as a property of projective varieties, since in $\mathbb{C}^n$, a linear space is not an intrinsic property. For example, in two variables, $x=0$ and $x=y^2$ define indistinguishable varieties, but the degrees of their equations are different. On the other hand, in $\mathbb{P}^n$ varieties are defined as zeroes of homogeneous polynomials. In particular, one would like to say that a hypersurface defined by a homogeneous polynomial of degree $d$ has degree $d$ as a variety. This can be translated as, if we intersect the hypersurface with a 'general' line, we should get $d$ points. So, the degree of an arbitrary projective variety of dimension $r$ in $\mathbb{P}^n$ is the number points in the intersection of the variety with a general linear subspace of dimension $n-r$. That this intersection is in fact a finite number of points and independent of the linear space as long as it is general are somewhat technical theorems. Of course, take some of this with a pinch of salt.