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

Source abstract

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 $n=1$.