We show that vanishing of asymptotic p-th syzygies implies p-very ampleness for line bundles on arbitrary projective schemes. For smooth surfaces we prove that the converse holds when p is small, by studying the Bridgeland-King-Reid-Haiman correspondence for tautological bundles on the Hilbert scheme of points. This extends previous results of Ein-Lazarsfeld, Ein-Lazarsfeld-Yang and gives a partial answer to some of their questions. As an application of our results, we show how to use syzygies to bound the irrationality of a variety.

中文翻译：

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渐近合子和高阶嵌入

我们表明渐近 p-th syzygies 的消失意味着在任意投影方案上线丛的 p-非常充足。对于光滑表面，我们通过研究 Hilbert 点方案上的重言丛的 Bridgeland-King-Reid-Haiman 对应关系，证明了当 p 较小时逆向成立。这扩展了 Ein-Lazarsfeld、Ein-Lazarsfeld-Yang 之前的结果，并部分回答了他们的一些问题。作为我们结果的应用，我们展示了如何使用 syzygies 来限制多样性的非理性。