We prove bounds on the saturation degrees of homogeneous ideals (and their powers) defining smooth complex projective varieties. For example, we show that a classical statement due to Macaulay for zero-dimensional complete intersection ideals holds for any smooth variety. For curves, we bound the saturation degree of powers in terms of the regularity.

中文翻译：

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平滑品种的饱和界限

我们证明了定义平滑复射影簇的齐次理想（及其幂）饱和度的界限。例如，我们证明了由于 Macaulay 的零维完全交集理想的经典陈述适用于任何平滑变体。对于曲线，我们根据规律性限制了功率的饱和度。