Log-concavity and unimodality of Hodge numbers of Hilbert schemes of points over a surface

Let \(S\) be a smooth projective complex surface with irregularity \(q=h^{1,0}(S)\) and geometric genus \(g=h^{2,0}(S)\), and let \(S^{[n]}\) denote its Hilbert scheme of \(n\) points. We prove that, for every \(n\ge0\), the sequence \[ \left(h^{p,0}\bigl(S^{[n]}\bigr)\right)_{p=0}^{2n} \] is log-concave if and only if \(g\le\binom{q+1}{2}\). Moreover, log-concavity of all these sequences is already equivalent to log-concavity of the sequence for \(n=2\). We also prove that these sequences are unimodal for every \(n\ge0\) if and only if \(q\ge1\) or \(g=0\), and that this condition is already detected by the sequence for \(n=1\).