Teixidor i Bigas, Montserrat Green’s conjecture for the generic \(r\)-gonal curve of genus \(g\geq 3r-7\). (English) Zbl 1059.14039 Duke Math. J. 111, No. 2, 195-222 (2002). Summary: The syzygies of a generic canonical curve are expected to be as simple as possible for \(p\leq(g- 3)/2\). We prove this result here for \(p\leq(g -2)/3\) only. The proof is carried out by considering infinitesimal deformations near a hyperelliptic curve. Cited in 3 ReviewsCited in 8 Documents MSC: 14H51 Special divisors on curves (gonality, Brill-Noether theory) 13D02 Syzygies, resolutions, complexes and commutative rings Keywords:syzygies; deformations × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] E. Arbarello and M. Cornalba, Su una congettura di Petri , Comment. Math. Helv. 56 (1981), 1–38. · Zbl 0505.14002 · doi:10.1007/BF02566195 [2] D. Bayer and D. Eisenbud, Ribbons and their canonical embeddings , Trans. Amer. Math. Soc. 347 (1995), 719–756. JSTOR: · Zbl 0853.14016 · doi:10.2307/2154871 [3] I. Biswas and S. Ramanan, An infinitesimal study of the moduli of Hitchin pairs , J. London Math. Soc. (2) 49 (1994), 219–231. · Zbl 0819.58007 · doi:10.1112/jlms/49.2.219 [4] L. Ein, A remark on the syzygies of the generic canonical curves , J. Differential Geom. 26 (1987), 361–365. · Zbl 0632.14024 [5] D. Eisenbud and M. Green, Clifford indices of ribbons , Trans. Amer. Math. Soc. 347 (1995), 757–765. JSTOR: · Zbl 0854.14016 · doi:10.2307/2154872 [6] W. Fulton and J. Harris, Representation Theory: A First Course , Grad. Texts in Math. 129 , Springer, New York, 1991. · Zbl 0744.22001 [7] M. Green, Koszul cohomology and the geometry of projective varieties , J. Differential Geom. 19 (1984), 125–171. · Zbl 0559.14008 [8] A. Hirschowitz and S. Ramanan, New evidence for Green’s Conjecture on syzygies of canonical curves , Ann. Sci. École Norm. Sup. (4) 31 (1998), 145–152. · Zbl 0923.14017 · doi:10.1016/S0012-9593(98)80013-X [9] K. Hulek, K. Paranjape, and S. Ramanan, On a conjecture on canonical curves , J. Algebraic Geom. 1 (1992), 335–359. · Zbl 0790.14031 [10] K. Paranjape and S. Ramanan, “On the canonical ring of a curve” in Algebraic Geometry and Commutative Algebra, II , Kinokuniya, Tokyo, 1988, 503–516. · Zbl 0699.14041 [11] F.-O. Schreyer, Syzygies of canonical curves and special linear series , Math. Ann. 275 (1986), 105–137. · Zbl 0578.14002 · doi:10.1007/BF01458587 [12] –. –. –. –., “Green’s conjecture for general \(p\)-gonal curves of large genus” in Algebraic Curves and Projective Geometry (Trento, Italy, 1988) , Lecture Notes in Math. 1389 , Springer, Berlin, 1989, 254–260. · Zbl 0691.14023 · doi:10.1007/BFb0085937 [13] –. –. –. –., A standard basis approach to syzygies of canonical curves , J. Reine Angew. Math. 421 (1991), 83–123. · Zbl 0578.14002 · doi:10.1007/BF01458587 [14] M. Teixidor i Bigas, Half-canonical series on algebraic curves , Trans. Amer. Math. Soc. 302 (1987), 99–115. JSTOR: · Zbl 0627.14022 · doi:10.2307/2000899 [15] C. Voisin, Courbes tétragonales et cohomologie de Koszul , J. Reine Angew. Math. 387 (1988), 111–121. · Zbl 0652.14012 · doi:10.1515/crll.1988.387.111 [16] G. E. Welters, Polarized abelian varieties and the heat equations , Compositio Math. 49 (1983), 173–194. · Zbl 0576.14042 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.