Dear Jason, is it correct to say that your argument works for projective spaces (or even smooth quadrics) of any dimension, if one assumes that singularities of the ramification are good enough (say, double points)? Of course $U$ would just be an open set whose complement is at least codimension two.

It is the Segre classes of $X$ inside $\mathbb{P}^n$, so I think they are the classes of the normal cone of $X$.

@Sasha: thank you, which is exactly the paper that you mention?

sorry Olivier, I might not have been clear. I am only talking about the case when $X$ is the moduli space.

sorry, I meant: Thank you, Jason. Yes in fact that's what I knew. In the case of stable bundles with trivial determinant the variety is still smooth and unirational, but just quasi-projective...

Yes in fact that's what I knew