We introduce a relativization of the secant sheaves from Green and Lazarsfeld (A simple proof of Petri’s theorem on canonical curves, Geometry Today, 1984) and Ein and Lazarsfeld (Inventiones Math 190:603-646, 2012) and apply this construction to the study of syzygies of canonical curves. As a first application, we give a simpler proof of Voisin’s Theorem for general canonical curves. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves. Secondly, in the case of curves of even genus, we enhance Voisin’s Theorem by providing a structure theorem for the last syzygy space, resolving the Geometric Syzygy Conjecture in even genus.

中文翻译：

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正则曲线的万有割丛和合子

我们介绍了 Green 和 Lazarsfeld（Petri's theorem on canonical curve, Geometry Today, 1984）和 Ein and Lazarsfeld（Inventiones Math 190:603-646, 2012）的割线轮的相对化，并将这种构造应用于研究规范曲线的合子。作为第一个应用，我们给出了一般典型曲线的 Voisin 定理的更简单的证明。这完全确定了此类曲线的坐标环的最小自由分辨率项。其次，在偶数属曲线的情况下，我们通过为最后一个合子空间提供结构定理来增强Voisin定理，解决偶数属中的几何合子猜想。