  Manin's conjecture predicts the number of rational points of bounded height
on a Fano variety. To make this prediction precise, it is necessary to remove a
thin subset of rational points. Peyre has tentatively proposed replacing this
subset by the set of points where a certain freeness function he defined takes
small values. We show that this proposal fails in the case of
$\operatorname{Hilb}^2(\mathbb P^n)$, because the usual thin subset, consisting
of rational points that lift to a certain double cover, contains many points
with relatively large freeness.
