Clifford indices of ribbons

Authors:
David Eisenbud and Mark Green

Journal:
Trans. Amer. Math. Soc. **347** (1995), 757-765

MSC:
Primary 14H45; Secondary 14C20

MathSciNet review:
1273474

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Abstract: We present a theory of "limit linear series" for rational ribbons-- that is, for schemes that are double structures on . This allows us to define a "linear series Clifford index" for ribbons. Our main theorem shows that this is the same as the Clifford index of ribbons studied by Eisenbud-Bayer in this same volume. This allows us to prove that the Clifford index is semicontinuous in degenerations from a smooth curve to a ribbon. A result of Fong [1993] then shows that ribbons may be deformed to smooth curves of the same Clifford index. Thus the Canonical Curve Conjecture of Green [1984] would follow, at least for a general smooth curve of each Clifford index, from the corresponding statement for ribbons.

**[D]**Dave Bayer and David Eisenbud,*Ribbons and their canonical embeddings*, Trans. Amer. Math. Soc.**347**(1995), no. 3, 719–756. MR**1273472**, 10.1090/S0002-9947-1995-1273472-3**[C]**Cyril D’Souza,*Compactification of generalised Jacobians*, Proc. Indian Acad. Sci. Sect. A Math. Sci.**88**(1979), no. 5, 419–457. MR**569548****[D]**David Eisenbud,*Green’s conjecture: an orientation for algebraists*, Free resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990) Res. Notes Math., vol. 2, Jones and Bartlett, Boston, MA, 1992, pp. 51–78. MR**1165318****[L]**Lung-Ying Fong,*Rational ribbons and deformation of hyperelliptic curves*, J. Algebraic Geom.**2**(1993), no. 2, 295–307. MR**1203687****[M]**Mark L. Green,*Koszul cohomology and the geometry of projective varieties*, J. Differential Geom.**19**(1984), no. 1, 125–171. MR**739785****1.**Jürgen Herzog,*Canonical Koszul cycles*, International Seminar on Algebra and its Applications (Spanish) (México City, 1991) Aportaciones Mat. Notas Investigación, vol. 6, Soc. Mat. Mexicana, México, 1992, pp. 33–41. MR**1310371**

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1995-1273474-7

Article copyright:
© Copyright 1995
American Mathematical Society