Let X be a smooth variety over a field k. The index of X over k is the gcd of the degrees [κ(x) : k] over all closed points x of X. The index is 1 if and only if X has a zero cycle of degree 1. If k is perfect, then the index of X is a birational invariant on smooth varieties over k: The reason is that given a nonempty open U of X and a closed point x in X you can find a curve C ⊂ X with x ∈ C, and it is easy to move zero cycles on curves. (I think the birational invariance also holds over nonperfect fields, but I haven’t checked this.)
Another birational invariant of a d-dimensional variety X over k is the gcd of the degrees of rational maps X —> P^d_k. This is the same as the gcd of closed subvarieties of P^n (any n) birational to X. Let’s temporarily call this the b-index. Note that by taking inverse images of k-rational points on P^d_k we see that index | b-index for smooth X (if k is finite you have to look at points over finite extensions). I claim that in fact index = b-index at least over a perfect field. After shrinking X we may assume that X is affine, hence quasi-projective, so X ⊂ P^N_k for some N >> 0 having some (super large and super divisible) degree D. On the other hand, consider the blow up b : X’ —> X of X in x. Then the invertible sheaf b^*O_X(N)(-Exceptional) will be very ample and will embed X’ into a large projective space where it has degree N^dD – [κ(x) : k]. This implies that b-index divides [κ(x) : k] and we win.