Abstract
Green’s canonical syzygy conjecture asserts a simple relationship between the Clifford index of a smooth projective curve and the shape of the minimal free resolution of its homogeneous ideal in the canonical embedding. We prove the analogue of this conjecture formulated by Bayer and Eisenbud for a class of non-reduced curves called ribbons. Our proof uses the results of Voisin and Hirschowitz–Ramanan on Green’s conjecture for general smooth curves.
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References
Aprodu, M.: Remarks on syzygies of \(d\)-gonal curves. Math. Res. Lett. 12(2–3), 387–400 (2005). doi:10.4310/MRL.2005.v12.n3.a9
Aprodu, M., Farkas, G.: Green’s conjecture for curves on arbitrary \(K3\) surfaces. Compos. Math. 147(3), 839–851 (2011). doi:10.1112/S0010437X10005099
Aprodu, M., Farkas, G.: Green’s conjecture for general covers. In: Alexeev, V., Gibney, A., Izadi, E., Kollár, J., Looijenga, E. (eds.) Compact moduli spaces and vector bundles, contemporary mathematics, vol. 564, pp. 211–226. American Mathematical Society, Providence (2012). doi:10.1090/conm/564/11147
Aprodu, M., Pacienza, G.: The Green conjecture for exceptional curves on a \(K3\) surface. Int. Math. Res. Not. IMRN 14(rnn043), 25 (2008). doi:10.1093/imrn/rnn043
Bayer, D., Eisenbud, D.: Ribbons and their canonical embeddings. Trans. AMS 347(3), 719–756 (1995)
Deopurkar, A., Fedorchuk, M., Swinarski, D.: Toward GIT stability of syzygies of canonical curves. Algebr. Geom. 3(1), 1–22 (2016). doi:10.14231/AG-2016-001
Ein, L., Lazarsfeld, R.: Asymptotic syzygies of algebraic varieties. Invent. Math. 190, 603–646 (2012). doi:10.1007/s00222-012-0384-5
Eisenbud, D., Green, M.: Clifford indices of ribbons. Trans. Amer. Math. Soc. 347(3), 757–765 (1995). doi:10.2307/2154872
Fong, L.Y.: Rational ribbons and deformation of hyperelliptic curves. J. Algebraic Geom. 2(2), 295–307 (1993)
Green, M., Lazarsfeld, R.: On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83(1), 73–90 (1986). doi:10.1007/BF01388754
Green, M.L.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19(1), 125–171 (1984)
Hall, J.: Moduli of singular curves. arXiv:1011.6007 [math.AG] (2010)
Hartshorne, R.: Deformation Theory, Graduate Texts in Mathematics, vol. 257. Springer, New York (2010). doi:10.1007/978-1-4419-1596-2
Hirschowitz, A., Ramanan, S.: New evidence for Green’s conjecture on syzygies of canonical curves. Ann. Sci. École Norm. Sup. (4) 31(2), 145–152 (1998). doi:10.1016/S0012-9593(98)80013-X
Lelli-Chiesa, M.: Green’s conjecture for curves on rational surfaces with an anticanonical pencil. Math. Z. 275(3–4), 899–910 (2013). doi:10.1007/s00209-013-1164-7
Schreyer, F.O.: Green’s conjecture for general \(p\)-gonal curves of large genus. In: Algebraic Curves and Projective Geometry (Trento, 1988), Lecture Notes in Math., vol. 1389, pp. 254–260. Springer, Berlin (1989). doi:10.1007/BFb0085937
Teixidor, I., Bigas, M.: Green’s conjecture for the generic \(r\)-gonal curve of genus \(g\ge 3r-7\). Duke Math. J. 111(2), 195–222 (2000). doi:10.1215/S0012-7094-02-11121-1
Voisin, C.: Green’s generic syzygy conjecture for curves of even genus lying on a \(K3\) surface. J. Eur. Math. Soc. (JEMS) 4(4), 363–404 (2002). doi:10.1007/s100970200042
Voisin, C.: Green’s canonical syzygy conjecture for generic curves of odd genus. Compos. Math. 141(5), 1163–1190 (2005). doi:10.1112/S0010437X05001387
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Deopurkar, A. The canonical syzygy conjecture for ribbons. Math. Z. 288, 1157–1164 (2018). https://doi.org/10.1007/s00209-017-1930-z
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DOI: https://doi.org/10.1007/s00209-017-1930-z