Abstract
We propose a new method, using deformation theory, to study the maximal rank conjecture. For line bundles of extremal degree, which can be viewed as the first case to test the conjecture, we prove that maximal rank conjecture holds by our new method.
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Arbarello E., Cornalba M.: Su una congettura di Petri. Comment. Math. Helvetici 56, 1–38 (1981)
Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of Algebraic Curves, vol. I. Springer Grundlehren 267 (1985)
Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Special Divisors on Algebraic Curves, vol. I. Regional Algebraic Geometry Conference, Athens, Georgia, May 1979
Ballico E., Ellia P.: The maximal rank conjecture for nonspecial curves in \({\mathbb{P}^3}\). Invent. Math 79, 541–555 (1985)
Ballico E., Ellia P.: On postulation of curves in \({\mathbb{P}^4}\). Math. Z. 188, 215–223 (1985)
Ballico E., Ellia P.: The maximal rank conjecture for nonspecial curves in \({\mathbb{P}^n}\). Math. Z. 196, 355–367 (1987)
Clemens H.: A local proof of Petri’s conjecture at the general curve. J. Differ. Geom. 54, 139–176 (2000)
Cukierman F., Ulmer D.: Curves of genus ten on K3 surfaces. Compos. Math. 89, 81–90 (1993)
Eisenbud D., Harris J.: A simpler proof of the Gieseker-Petri theorem on special divisors. Invent. Math 74, 269–280 (1983)
Eisenbud D., Harris J.: Divisors on general curves and cuspidal rational curves. Invent. Math. 74, 371–418 (1983)
Eisenbud D., Harris J.: Limit linear series: basic theory. Invent. Math. 85, 337–371 (1986)
Eisenbud D., Harris J.: Irreducibility and monodromy of some families of linear series. Annales scientifiques de l’École Normale Supérieure Sér. 4(20), 65–87 (1987)
Farkas G., Popa M.: Effective divisors on \({\mathcal{M}_g}\), curves on K3 surfaces, and the slope conjecture. J. Algebraic Geom. 14(2), 241–267 (2005)
Gieseker D.: Stable curves and special divisors. Invent. Math. 66, 251–275 (1982)
Green M.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19, 125–171 (1984)
Griffiths P., Harris J.: On the variety of linear systems on a general algebraic curve. Duke Math. J. 47, 233–272 (1980)
Green M., Lazarsfeld R.: On the projective normality of complete linear series on an algebraic curve. Invent. Math. 83, 73–90 (1986)
Green M., Lazarsfeld R.: Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville. Invent. Math. 90, 389–407 (1987)
Harris, J. Curves in projective space. Les Press de l’Université de Montréal (1982)
Harris J., Morrison I.: Moduli of curves. Graduate Text in Mathematics, vol. 187. Springer, Berlin (1998)
Lehman, R.: Brill-Noether type theorems with a movable ramification point. Ph.D thesis, MIT. http://dspace.mit.edu/handle/1721.1/38885 (2007)
Mumford, D.: Varieties defined by quadratic equations. C.I.M.E. Conference on Questions on Algebraic Varieties, pp. 31–100 (1969)
Osserman B.: The number of linear series on curves with given ramification. Int. Math. Res. Not. 47, 2513–2527 (2003)
Sernesi, E.: Deformations of Algebraic Schemes. Springer Grundlehren 334 (2006)
Wang, J.: Deformations of pairs (X, L) when X is singular. Proc. A.M.S. (to appear) (arxiv.org/abs/1003.6073)
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Wang, J. On the projective normality of line bundles of extremal degree. Math. Ann. 355, 1007–1024 (2013). https://doi.org/10.1007/s00208-012-0809-x
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DOI: https://doi.org/10.1007/s00208-012-0809-x