皮卡模簇的环面紧化上的有理曲线
Rational curves on toroidal compactifications of Picard modular varieties
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中文总结 AI 辅助
该研究改进了皮卡模簇环面紧化典范丛丰富性的维数界，构造了含特定有理曲线的无穷多不可公度非紧皮卡模簇，证明相关曲线在任意高维均存在。
中文摘要 AI 辅助
已知非紧有限体积球商的光滑环面紧化的典范丛在维数n≥3时是数值有效的（nef），在n≥6时是丰富的（ample）。我们证明该典范丛在n≥4时是丰富的，并表明该维数界是紧的。我们构造了无穷多个两两不可公度的非紧皮卡模三维簇，其光滑射影环面紧化具有非丰富的典范丛，每个紧化都包含一条与内部相交且典范次数为零的光滑有理曲线。更一般地，在每个维数n≥2中，我们构造了无穷多个两两不可公度的非紧皮卡模簇，其光滑环面紧化包含与内部相交的光滑有理曲线，特别是这类曲线在任意高维数下都存在，即使典范丛是丰富的。
英文摘要
It was known that the canonical bundle of a smooth toroidal compactification of a noncompact finite-volume ball quotient is nef in dimension $n\geq3$ and ample in dimension $n\geq6$. We prove that the canonical bundle is ample in dimension $n\geq4$ and show that this dimension bound is sharp. We construct infinitely many pairwise noncommensurable noncompact Picard modular threefolds whose smooth projective toroidal compactifications have non-ample canonical bundle. Each compactification contains a smooth rational curve meeting the interior and having canonical degree zero. More generally, in every dimension $n\geq2$, we construct infinitely many pairwise noncommensurable noncompact Picard modular varieties whose smooth toroidal compactifications contain smooth rational curves meeting the interior. In particular, such curves persist in arbitrarily high dimension, even when the canonical bundle is ample.
