The Gaussian map for rational ruled surfaces

Authors:
Jeanne Duflot and Rick Miranda

Journal:
Trans. Amer. Math. Soc. **330** (1992), 447-459

MSC:
Primary 14J26; Secondary 14E25, 14H99

MathSciNet review:
1061775

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper the Gaussian map of a smooth curve lying on a minimal rational ruled surface is computed. It is shown that the corank of is determined for almost all such curves by the rational surface in which it lies. Hence, except for some special cases, a curve cannot lie on two nonisomorphic minimal rational ruled surfaces.

**[B-M]**A. Beauville and J.-Y. Mérindol,*Sections hyperplanes des surfaces 𝐾3*, Duke Math. J.**55**(1987), no. 4, 873–878 (French). MR**916124**, 10.1215/S0012-7094-87-05541-4**[C-H-M]**Ciro Ciliberto, Joe Harris, and Rick Miranda,*On the surjectivity of the Wahl map*, Duke Math. J.**57**(1988), no. 3, 829–858. MR**975124**, 10.1215/S0012-7094-88-05737-7**[C-M1]**Ciro Ciliberto and Rick Miranda,*On the Gaussian map for canonical curves of low genus*, Duke Math. J.**61**(1990), no. 2, 417–443. MR**1074304**, 10.1215/S0012-7094-90-06118-6**[C-M2]**-,*Gaussian maps for certain families of canonical curves*, Proc. Bergen 1989 Conf. in Algebraic Geometry (to appear).**[G-H]**Phillip Griffiths and Joseph Harris,*Principles of algebraic geometry*, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR**507725****[H]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[M]**Rick Miranda,*The Gaussian map for certain planar graph curves*, Algebraic geometry: Sundance 1988, Contemp. Math., vol. 116, Amer. Math. Soc., Providence, RI, 1991, pp. 115–124. MR**1108635**, 10.1090/conm/116/1108635**[S-D]**B. Saint-Donat,*Projective models of 𝐾-3 surfaces*, Amer. J. Math.**96**(1974), 602–639. MR**0364263****[W1]**Jonathan M. Wahl,*The Jacobian algebra of a graded Gorenstein singularity*, Duke Math. J.**55**(1987), no. 4, 843–871. MR**916123**, 10.1215/S0012-7094-87-05540-2**[W2]**-,*Gaussian maps on algebraic curves*, Preprint.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
14J26,
14E25,
14H99

Retrieve articles in all journals with MSC: 14J26, 14E25, 14H99

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1061775-4

Article copyright:
© Copyright 1992
American Mathematical Society