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Universal secant bundles and syzygies of canonical curves

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Abstract

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.

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Acknowledgements

I thank C. Voisin for helpful explanations and G. Farkas for numerous discussions. I thank R. Lazarsfeld for encouragement and for detailed comments. I thank M. Aprodu, J. Ellenberg, D. Erman, D. Huybrechts, J. Rathmann, E. Sernesi and R. Yang for feedback on previous versions. I thank the referee for a careful reading and for comments which greatly improved the exposition. The author is supported by NSF grant DMS-1701245.

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  1. Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr, Madison, WI, 53706, USA

    Michael Kemeny

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Kemeny, M. Universal secant bundles and syzygies of canonical curves. Invent. math. 223, 995–1026 (2021). https://doi.org/10.1007/s00222-020-01001-5

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