Abstract
Let S⊂G(1,3)⊂p5 be a smooth, irreducible, non degenerate surface in the complex grassmannian G(1,3). Assume deg(S)=9, we show that S is one of the following surfaces:
-
(a)
A K3 surface blown up in one point.
-
(b)
The image of P2 by the linear system\(\left| {O_{P^2 } (6) - 2b_1 - \ldots - 2b_5 - b_6 - \ldots - b_{12} } \right|\)
-
(c)
The image of P2 by the linear system\(\left| {O_{P^2 } (7) - 2b_1 - \ldots - 2b_{10} } \right|\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
ARBARELLO E., CORNALBA M., GRIFFITHS P., HARRIS J.: Geometry of Algebraic Curves I, Berlin-Heidelberg-New York, Springer 1984
ARRONDO E., Sols I.: Classification of smooth congruences of low degree, preprint, 1987
BARTH W., PETERS C., VAN De VEN A.: Compact complex surfaces, Berlin-Heidelberg-New York, Springer 1984
BEAUVILLE A.: Surfaces Algebriques Complexes, Astérisque54, 1978
KLEIMAN S.: The transversality of a general translate, Comp. Math.28, 287–297 1974
BORDIGA G.: Di una certa congruenza del terzo ordine e della sesta classe dello spazio ordinario, Rend. Ace. Lincei4, 8–13 1890
EIN L.: Surfaces with a hyperelliptic hyperplane section, Duke math. J.50, 685–694, 1984
FANO G.: Studio di alcuni sistemi di rette considerati come superficie dello spazio a cinque dimensioni, Ann. Mat. pura appl.21, 141–192, 1893
GRIFFITHS P., HARRIS J.: Priciples of Algebraic Geometry, New York, Wiley and S., 1978
HARTSHORNE R.: Algebraic Geometry, Berlin-Heidelberg-New York, Springer 1977
LIVORNI E.L.: Classification of algebraic non ruled surfaces with sectional genus less or equal to six, Nagoya Math. J.100, 1–9, 1985
LIVORNI E.L.: Classification of algebraic surfaces with sectional genus less or equal to six: Rational surfaces, Pac. J. of Math.113, 93–113, 1984
LIVORNI E.L.: Classification of algebraic surfaces with sectional genus less or equal to six; ruled surfaces with dim ΦK⊗L(X)=2, Math. Scand.58, 9–29, 1986
LIVORNI E.L.: Classification of algebraic surfaces with sectional genus less or equal to six; ruled surfaces with dim ΦK⊗L(X)=1, Can. J. Math.37, 1110–1121, 1986
OKONEK C.: Flaechen vom grad 8 in P4, Math. Zeits.191, 207–223, 1986
OKONEK C.: Moduli reflexiver Garben und Flaechen vom Kleinem Grad in P4, Math. Zeits.184, 549–572, 1983
OKONEK C.: Ueber 2-codimensionale Untermannigfaltigkeiten vom Grad 7 in P4 und P5, Math. Zeits.187, 209–219, 1984
PAPANTONOPOULOU A.: Embeddings in G(1,3), Proc. A.M.S.89,583–586, 1983
RAN Z.: Surfaces of order 1 in Grassmannians, J.f.d. Reine u. Angew. Mathematik362, 1986
REIDER I.: Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math.127, 309–316, 1988
SAINT DONAT D.: Projective models of K3 surfaces, Am. J. Math.96, 602–639, 1974
SOMMESE A.J.: Hyperplane sections of projective surfaces I - The Adjunction mapping, Duke Math. J.46, 377–401, 1979
SOMMESE A.J., VAN DE VEN A.: On the Adjunction mapping, Math. Ann.278, 593–603, 1987
VAN DE VEN A.: On the 2-connectedness of very ample divisors on a surface, Duke Math. J.46, 403–407, 1979
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Verra, A. Smooth surfaces of degree 9 in G(1,3). Manuscripta Math 62, 417–435 (1988). https://doi.org/10.1007/BF01357719
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01357719