  Let G be one of the groups SL_n C, Sp_2n C, SO_m C, O_m C, or G_2. For a
generically free G-representation V, we say that N is a level of stable
rationality for V/G if V/G x P^N is rational. In this paper we improve known
bounds for the levels of stable rationality for the quotients V/G. In
particular, their growth as functions of the rank of the group is linear for G
one of the classical groups.
