Consider all the smooth hypersurfaces of degree $d$ in $\mathbb P^n$, for $d,n$ general enough. If $X$ and $Y$ are birational equivalent, then I think they are not necessarily isomorphic. I want to know that:
what is the dimension of birational equivalent classes of hypersurfaces?
For example, for $(n,d)=(3,3)$, all smooth cubic surfaces are birational equivalent, so the dimension is $0$. I would like to know how should we approach this in general.