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Freeness alone is insufficient for Manin-Peyre
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abstract
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.
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Cited by 1 Pith paper
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Manin's conjecture for the chordal cubic fourfold
The number of rational points of bounded anticanonical height on the desingularized chordal cubic fourfold, outside the expected thin set, matches Manin's conjecture exactly.
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