发表机构
IIT Madras（印度理工学院马德拉斯分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对光滑复射影曲面的n点Hilbert概型，证明了其Hodge数序列的对数凹性等价于g≤组合数且由n=2的序列决定，单峰性等价于q≥1或g=0且由n=1的序列判定。
AI 中文摘要
设S是光滑复射影曲面，非正则性q=h^{1,0}(S)，几何亏格g=h^{2,0}(S)，记S^{[n]}为其n个点的Hilbert概型。我们证明：对任意n≥0，序列(h^{p,0}(S^{[n]}))_{p=0}^{2n}为对数凹序列当且仅当g≤\inom{q+1}{2}，且所有这类序列的对数凹性等价于n=2时对应序列的对数凹性。我们还证明：对任意n≥0，这些序列为单峰序列当且仅当q≥1或g=0，且该条件可由n=1时的序列判定。
英文摘要
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 \(n=1\).
CommentsThis is a preliminary version. Comments are welcome
