Koszul modules and Green’s conjecture. (English) Zbl 1430.14074
M. L. Green’s conjecture [J. Differ. Geom. 19, 125–167, 168–171 (1984; Zbl 0559.14008)] predicts the expected (non-)vanishing behavior for the syzygies of canonical curves. For general canonical curves the conjecture was established in characteristic zero by C. Voisin [J. Eur. Math. Soc. (JEMS) 4, No. 4, 363–404 (2002; Zbl 1080.14525); Compos. Math. 141, No. 5, 1163–1190 (2005; Zbl 1083.14038)] using \(K3\) surfaces and their Hilbert schemes of points. In this paper, the authors take a different approach by studying the syzygies of the tangent surface to a rational normal curve in \(\mathbb{P}^g\), whose hyperplane section gives a rational \(g\)-cuspidal curve as the degeneration of canonical curves of genus \(g\). As a consequence, the authors prove that Green’s conjecture holds for general canonical curves in characteristic zero as well as in positive characteristic \(\mathrm{char}(\mathbf{k})\geq \frac{g+2}{2}\). In order to prove the result, the authors describe an explicit, characteristic-independent version of Hermite reciprocity for \(\mathfrak{sl}_2\)-representations. Moreover, they give a complete description in arbitrary characteristics for the (non-)vanishing behavior of the syzygies of the tangential surface to a rational normal curve.
Reviewer: Dawei Chen (Chestnut Hill)
MSC:
| 14H99 | Curves in algebraic geometry |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
Software:
K3CarpetsReferences:
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