Ribbons and their canonical embeddings

Authors:
Dave Bayer and David Eisenbud

Journal:
Trans. Amer. Math. Soc. **347** (1995), 719-756

MSC:
Primary 14H45; Secondary 13D02, 14C20

DOI:
https://doi.org/10.1090/S0002-9947-1995-1273472-3

MathSciNet review:
1273472

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study double structures on the projective line and on certain other varieties, with a view to having a nice family of degenerations of curves and K3 surfaces of given genus and Clifford index. Our main interest is in the canonical embeddings of these objects, with a view toward Green's Conjecture on the free resolutions of canonical curves. We give the canonical embeddings explicitly, and exhibit an approach to determining a minimal free resolution.

**[D]**Bayer and M. Stillman,*Macaulay: A system for computation in algebraic geometry and commutative algebra*, Available from the authors or by anonymous ftp as follows. ftp zariski.harvard.edu, login: anonymous, Password: any, cd Macaulay, binary, get M3.tar, quit, tar xf M3.tar..**[K]**Karen A. Chandler,*Geometry of dots and ropes*, Trans. Amer. Math. Soc.**347**(1995), no. 3, 767–784. MR**1273473**, https://doi.org/10.1090/S0002-9947-1995-1273473-5**[D]**David A. Buchsbaum and David Eisenbud,*What makes a complex exact?*, J. Algebra**25**(1973), 259–268. MR**0314819**, https://doi.org/10.1016/0021-8693(73)90044-6**[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****[D]**David Eisenbud and Mark Green,*Clifford indices of ribbons*, Trans. Amer. Math. Soc.**347**(1995), no. 3, 757–765. MR**1273474**, https://doi.org/10.1090/S0002-9947-1995-1273474-7**[D]**David Eisenbud and Joe Harris,*Finite projective schemes in linearly general position*, J. Algebraic Geom.**1**(1992), no. 1, 15–30. MR**1129837****1.**-,*On varieties of minimal degree*, Algebraic Geometry; Bowdoin 1985. Proc. Sympos. Pure Math., vol. 46, part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 3-14.**[L]**Lung-Ying Fong,*Rational ribbons and deformation of hyperelliptic curves*, J. Algebraic Geom.**2**(1993), no. 2, 295–307. MR**1203687****[L]**Laurent Gruson and Christian Peskine,*Courbes de l’espace projectif: variétés de sécantes*, Enumerative geometry and classical algebraic geometry (Nice, 1981), Progr. Math., vol. 24, Birkhäuser, Boston, Mass., 1982, pp. 1–31 (French). MR**685761****[R]**Robin Hartshorne,*Curves with high self-intersection on algebraic surfaces*, Inst. Hautes Études Sci. Publ. Math.**36**(1969), 111–125. MR**0266924****[K]**K. Hulek and A. Van de Ven,*The Horrocks-Mumford bundle and the Ferrand construction*, Manuscripta Math.**50**(1985), 313–335. MR**784147**, https://doi.org/10.1007/BF01168835**[S]**S. Lichtenbaum and M. Schlessinger,*The cotangent complex of a morphism*, Trans. Amer. Math. Soc.**128**(1967), 41–70. MR**0209339**, https://doi.org/10.1090/S0002-9947-1967-0209339-1**[R]**Ragni Piene,*Numerical characters of a curve in projective 𝑛-space*, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 475–495. MR**0506323****[F]**Frank-Olaf Schreyer,*Syzygies of canonical curves and special linear series*, Math. Ann.**275**(1986), no. 1, 105–137. MR**849058**, https://doi.org/10.1007/BF01458587**[E]**Edoardo Sernesi,*Topics on families of projective schemes*, Queen’s Papers in Pure and Applied Mathematics, vol. 73, Queen’s University, Kingston, ON, 1986. MR**869062**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
14H45,
13D02,
14C20

Retrieve articles in all journals with MSC: 14H45, 13D02, 14C20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1273472-3

Keywords:
Ribbon,
double structure,
hyperelliptic curve,
Clifford index,
Green's conjecture,
free resolution,
canonical curve,
K3 surface,
K3 carpet

Article copyright:
© Copyright 1995
American Mathematical Society