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Koszul modules and Green’s conjecture

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We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green’s conjecture for every g-cuspidal rational curve over an algebraically closed field \({{\mathbf {k}}}\), with \({\text {char}}({{\mathbf {k}}})=0\) or \({\text {char}}({{\mathbf {k}}})\ge \frac{g+2}{2}\). As a consequence, we deduce that the general canonical curve of genus g satisfies Green’s conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for \({{\mathfrak {s}}}{{\mathfrak {l}}}_2\)-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve.

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We acknowledge with thanks the contribution of A. Suciu. This project, including the companion paper [2], started with the paper [23], and since then we benefited from numerous discussions with him. We warmly thank A. Beauville, L. Ein, D. Eisenbud, B. Klingler, P. Pirola, F.-O. Schreyer and C. Voisin for interesting discussions related to this circle of ideas. We are particularly grateful to R. Lazarsfeld who read an early version of the paper and suggested many improvements which significantly clarified the exposition. Aprodu was partially supported by the Romanian Ministry of Research and Innovation, CNCS - UEFISCDI, grant PN-III-P4-ID-PCE-2016-0030, within PNCDI III. Farkas was supported by the DFG grant Syzygien und Moduli. Raicu was supported by the Alfred P. Sloan Foundation and by the NSF Grant No. 1600765. Weyman was partially supported by the Sidney Professorial Fund and the NSF grant No. 1802067.

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Aprodu, M., Farkas, G., Papadima, Ş. et al. Koszul modules and Green’s conjecture. Invent. math. 218, 657–720 (2019).

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